FAMILIES OF DIFFERENTIAL OPERATORS ACTING ON OVERCONVERGENT HILBERT MODULAR FORMS by JON AYCOCK A DISSERTATION Presented to the Department of Mathematics and the Division of Graduate Studies of the University of Oregon in partial fulfillment of the requirements for the degree of Doctor of Philosophy September 2022 DISSERTATION APPROVAL PAGE Student: Jon Aycock Title: Families of Differential Operators Acting on Overconvergent Hilbert Modular Forms This dissertation has been accepted and approved in partial fulfillment of the requirements for the Doctor of Philosophy degree in the Mathematics Department by: Ellen Eischen Chairperson Nicolas Addington Core Member Elena Mantovan Core Member Victor Ostrik Core Member Hank Childs Institutional Representative and Krista Chronister Vice Provost for Graduate Studies Original approval signatures are on file with the University of Oregon Division of Graduate Studies. Degree awarded September 2022 c 2022 Jon Aycock All rights reserved. ii DISSERTATION ABSTRACT Jon Aycock Doctor of Philosophy Department of Mathematics September 2022 Title: Families of Di↵erential Operators Acting on Overconvergent Hilbert Modular Forms We construct di↵erential operators acting on overconvergent Hilbert modular forms. This extends work of Katz in the case of p-adic Hilbert modular forms in [Kat78], and of Harron–Xiao and Liu for overconvergent Siegel modular forms in [HX14, Liu19a]. The result has applications to the construction of p-adic L- functions in the presence of a Damerell-type formula. iii CURRICULUM VITAE NAME OF AUTHOR: Jon Aycock GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED: University of Oregon, Eugene, OR, USA University of North Carolina, Chapel Hill, NC, USA DEGREES AWARDED: Doctor of Philosophy, Mathematics, 2022, University of Oregon Bachelor of Science, Mathematics, 2016, University of North Carolina AREAS OF SPECIAL INTEREST: Number Theory Automorphic Forms and their L-functions p-adic Avatars of Automorphic Forms PROFESSIONAL EXPERIENCE: Graduate Employee, Department of Mathematics, 2016-Present GRANTS, AWARDS AND HONORS: Alfred Brauer Award, University of North Carolina, 2016 Anderson Research Award, University of Oregon, 2020 College of Arts and Sciences Dissertation Research Fellowship, University of Oregon, 2020-2021 PUBLICATIONS: iv Families of Di↵erential Operators for Hilbert Modular Forms. Submitted Sep 2021. Preprint available at https://arxiv.org/abs/2006.02543 Categorifying Quadratic Zeta Functions (with Andrew Kobin). Preprint available at https://arxiv.org/abs/2205.06298 v ACKNOWLEDGEMENTS I would like to thank my advisor Ellen Eischen. There are many reasons why I wouldn’t be where I am right now if it weren’t for your mentorship and guidance. I also want to thank the rest of my committee including Nicolas Addington, Hank Childs, Elena Mantovan, and Victor Ostrik. For their mathematical input while I was writing the material for this dissertation, I would like to thank Giacomo Graziani, Sean Howe, and Zheng Liu. I would also like to thank the number theory and algebraic geometry group at Oregon, especially Corey Brooke, Maria Fox, Cathy Hsu, and Samantha Platt, for the sense of community you helped build, as well as the discussions we had about math. Also, I want to thank my ocemates Kelly Pohland and Elisha Wol↵, and the rest of my friends in the department, for all the time we’ve spent together on campus, both talking about math and otherwise. I would also like to mention the Rethinking Number Theory conference for working to create a framework in which mathematics in general, and number theory in particular, are more accessible to a wider array of people; and especially Andrew Kobin, for leading our project that began there. Finally, a large thank you to the rest of my friends and family back home and in Eugene, for supporting me and grounding me through my time in school. Special thanks to my mom, my dad, and my brothers for encouraging me to keep learning and always being interested in what I’m doing; to my friends that I’ve known so long that may as well be family; and to the ultimate frisbee community that welcomed me. vi TABLE OF CONTENTS Chapter Page I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2. Description of Results . . . . . . . . . . . . . . . . . . . . . 2 1.3. Relationship to Recent Developments . . . . . . . . . . . . . . 4 1.4. Notation and Conventions . . . . . . . . . . . . . . . . . . . 6 II. HILBERT MODULAR FORMS . . . . . . . . . . . . . . . . . . 7 2.1. Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2. Moduli Setup . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3. Hilbert Modular Forms . . . . . . . . . . . . . . . . . . . . . 9 2.4. Holomorphic Hilbert Modular Forms . . . . . . . . . . . . . . . 12 III. NEARLY HILBERT MODULAR FORMS . . . . . . . . . . . . . 19 3.1. The de Rham Sheaf . . . . . . . . . . . . . . . . . . . . . . 20 3.2. Associated Bundles . . . . . . . . . . . . . . . . . . . . . . 20 3.3. The Gauss–Manin Connection . . . . . . . . . . . . . . . . . 22 3.4. (g, Q)-modules . . . . . . . . . . . . . . . . . . . . . . . . 24 3.5. Nearly Hilbert Modular Forms . . . . . . . . . . . . . . . . . 27 3.6. Nearly Holomorphic Hilbert Modular Forms . . . . . . . . . . . 32 3.7. Splitting the Hodge Filtration . . . . . . . . . . . . . . . . . . 39 IV. P -ADIC THEORY . . . . . . . . . . . . . . . . . . . . . . . . 43 4.1. Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.2. Canonical Subgroups for Ordinary HBAVs . . . . . . . . . . . . 44 4.3. p-adic Hilbert Modular Forms . . . . . . . . . . . . . . . . . . 50 vii Chapter Page 4.4. Overconvergent Loci and Partial Canonical Subgroups . . . . . . . 52 4.5. Overconvergent Hilbert Modular Forms . . . . . . . . . . . . . 55 4.6. Nearly Overconvergent Hilbert Modular Forms and the Main Construction . . . . . . . . . . . . . . . . . . . . . . 57 4.7. Integrality . . . . . . . . . . . . . . . . . . . . . . . . . . 60 V. DESCENT TO G . . . . . . . . . . . . . . . . . . . . . . . . 63 5.1. Two Groups . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.2. Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 5.3. Descent of Modular Forms . . . . . . . . . . . . . . . . . . . 64 5.4. Descent of Operators . . . . . . . . . . . . . . . . . . . . . 65 APPENDIX: DAMERELL’S FORMULA . . . . . . . . . . . . . . . . 67 A.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 67 A.1.1. Notation, Conventions, and Assumed Knowledge . . . . . . 68 A.2. L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . 68 A.2.1. The 1-variable L-function . . . . . . . . . . . . . . . . 68 A.2.2. Hecke Characters . . . . . . . . . . . . . . . . . . . . 69 A.2.3. The (d+1)-variable L-function . . . . . . . . . . . . . . 70 A.3. Hilbert Modular Forms . . . . . . . . . . . . . . . . . . . . . 71 A.3.1. Setup . . . . . . . . . . . . . . . . . . . . . . . . . 72 A.3.2. Lattices . . . . . . . . . . . . . . . . . . . . . . . . 72 A.3.3. Holomorphic Functions . . . . . . . . . . . . . . . . . 74 A.3.4. Geometry . . . . . . . . . . . . . . . . . . . . . . . 75 A.4. Damerell’s Formula . . . . . . . . . . . . . . . . . . . . . . 78 A.4.1. Holomorphic Eisenstein Series . . . . . . . . . . . . . . 78 viii Chapter Page A.4.2. Maass–Shimura Operators . . . . . . . . . . . . . . . . 80 A.4.3. Nearly Holomorphic Eisenstein Series . . . . . . . . . . . 81 A.4.4. Damerell’s Formula . . . . . . . . . . . . . . . . . . . 82 A.5. Takeaways . . . . . . . . . . . . . . . . . . . . . . . . . . 83 A.5.1. Algebraicity of L-Values: Damerell’s Theorem . . . . . . . 84 A.5.2. p-adic Interpolation . . . . . . . . . . . . . . . . . . . 86 REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . 88 ix CHAPTER I INTRODUCTION In this thesis, we construct di↵erential operators acting on overconvergent Hilbert modular forms. This extends work of Katz in the case of p-adic Hilbert modular forms, and of Harron–Xiao and Liu for overconvergent Siegel modular forms. The result has applications to the construction of p-adic L-functions. 1.1 Motivation In [Ser72], Serre introduced the idea of using p-adic families of modular forms to p-adically interpolate values of L-functions. In particular, he used the family of Eisenstein series with q-expansion X1 2G2k(q) = ⇣(1 2 nk) + 2 2k1(n)q . n=1 He showed that congruences between the coecients of nq for n 1 for particular values of k imply the existence of such a congruence between the constant terms ⇣(1 k) as well, and used this to show that (an appropriate normalization of) the Riemann zeta function is p-adically continuous when restricted to inputs of negative odd integers. This is one construction of the Kubota–Leopoldt p-adic zeta function, which interpolates the values of the Riemann zeta function at negative odd integers. Serre was able to use a well-known family of modular forms whose values at the cusp 1 are equal to special values of the Riemann zeta function. Though the result was generalized to Dedekind zeta functions for totally real fields in [DR80], these families can be tricky to come up with in more generality. One important addition to the theory was Katz’s use of di↵erential operators, in particular those built from the Gauss–Manin connection r, as a key ingredient in the construction of these families in [Kat76, Kat78]. Katz’s operators give a p-adic analog of the 1 Maass–Shimura operators which Shimura used to prove algebraicity results in [Shi76, Shi00], and which were adapted by Harris to prove algebraicity results for higher rank groups in [Har81, Har86]. Katz leveraged this idea to p-adically interpolate the zeta function for a CM field K using Hilbert modular forms on its maximal totally real subfield +K . He started with a holomorphic Eisenstein series and used successive powers of these di↵erential operators to produce a family of not necessarily holomorphic Eisenstein series. By Damerell’s formula (in e.g. [GS81] and described in Appendix A), sums of the values of these Eisenstein series at the CM points of the modular curve give the central values L(, s0) of the L-functions for a specific class of Hecke characters , known as characters of type A0. Viewing the zeta function as a function on the space of characters, this fact reduces the p-adic interpolation of ⇣K to the study of how powers of the Gauss–Manin connection behave p-adically. As noted in his introduction, Katz’s method only succeeds when we can choose an “ordinary” CM type, which is only possible when every prime above p in the totally real field +K splits in +K/K . This is due to the fact that his di↵erential operators are only defined over the ordinary locus, which does not contain the CM points that are supersingular at p. The present work extends these operators to be defined on the overconvergent loci, using methods from [Liu19a] and using the geometry developed in [AIP15, AIP16]. This extension allows Damerell’s formula to be used in more general situations, whenever the Eisenstein series is defined at the CM points. 1.2 Description of Results The present work culminates in the following theorem, which is Theorem 4.6.2 in the body of the dissertation: 2 Theorem 1.2.1. Fix a tuple v = (vp)p|p with each vp > 0. For each embedding : F ! K, and any k 1, there is a di↵erential operator rk acting on families of nearly v-overconvergent Hilbert modular forms, which raises the weight by 2k and the type by k. The operators rk and r` commute for any pair of embeddings and ⌧ ⌧ . These rk are the p-adic analogs of the Maass-Shimura operators in the Archimedean case, as we prove in Section 3.5. The absence of these “overconvergent” operators in [Kat78] prevented Katz from constructing p-adic L-functions for CM fields in the case that the CM points were not ordinary; i.e., when +p is not split in K/K . The overconvergent operators for elliptic modular forms have been previously constructed, and are used in [AI19] to construct p-adic L-functions for quadratic imaginary fields; our construction lays the groundwork to generalize that construction to a general CM field. In defining the di↵erential operators, we work rigid analytically. However, we give a discussion of the integrality of the operator in Section 4.7. As in [AIP16], we are careful to make a distinction between two possible meanings of the phrase “Hilbert modular form.” Given a totally real field F , the term may refer to the automorphic forms on either of the following groups: ⇤ G = ResO /Z GL2, or G = G⇥F ResO G Gm.F /Z m For any commutative ring R, the R-points of G are the 2 ⇥ 2 invertible matrices with entries in OF ⌦Z R, while the R-points of ⇤G are those matrices from G(R) with determinant in ⇥ ⇥R ⇢ (OF ⌦Z R) . Each of these groups has an advantage: G has a nicer Hecke theory including a commutative Hecke algebra; while ⇤G allows us to use geometric tools, since its associated PEL-type moduli problem is representable by a Shimura variety. The inclusion ⇤G ⇢ G gives a restriction map 3 from the space of automorphic forms on G to those on ⇤G , and in fact there is an explicit geometric criterion that picks out the space of automorphic forms for G inside the space of automorphic forms for ⇤G . In the following, we focus first on ⇤ G so that we may use the geometric tools it a↵ords us, after which we shift our attention to this criterion which allows us to transport our results to the group G. We get the following theorem, Theorem 5.4.2 in the body of this dissertation, which allows these operators to be used in either situation: Theorem 1.2.2. The di↵erential operators rk constructed in Theorem 1.2.1 preserve the space of Hilbert modular forms for G inside the space of Hilbert modular forms for ⇤G . In Chapter 2, we give a geometric construction of Hilbert modular forms fort he group ⇤G . In Section 2.4, we give a comparison theorem that shows that our geometric Hilbert modular forms, when viewed over C, are equivalent to the usual notion of Hilbert modular forms in terms of holomorphic functions on a symmetric space. In Chapter 3, we repeat this for nearly Hilbert modular forms, including a description of a Maass–Shimura di↵erential operator in each setting, and a proof that they correspond. Chapter 4 then uses this as a basis to construct analogous spaces of p-adic Hilbert modular forms, and then overconvergent and nearly overconvergent Hilbert modular forms, as well as a Maass–Shimura di↵erential operator in this setting. Finally, in Chapter 5, we transport our results from ⇤G to G. 1.3 Relationship to Recent Developments This dissertation adds to the recent investigations into extending p-adic di↵erential operators past the ordinary locus, or into when the ordinary locus is empty, including those in [dSG14, dSG19, EM20, Urb14, Liu19a, Liu19b]. Shortly 4 after posting the paper on which this dissertation is based on the arXiv, the author learned from Giacomo Graziani that his dissertation [Gra20] (currently in preparation) is closely related. His dissertation uses the VBMS (vector bundles with marked sections) formalism present in e.g. [AI19], while we use (g, Q)-modules as in [Liu19a]. In [HX14], the authors ask whether or not their construction can be adapted to avoid a choice of the Hodge filtration. Building on the work of [AI19, Liu19a], this work answers that question in the armative. Additionally, other works note that in order to use nearly overconvergent Hilbert modular forms to interpolate the values of p-adic L-functions, a “canonical splitting of the Hodge filtration” must be chosen; see for example the introduction to [AI19] for this in the situation of Damerell’s formula. Our methods translate this into a need for a canonical choice of frame for the relative de Rham cohomology of the modular curve in the nearly holomorphic case. Seeing as a canonical trivialization (i.e., a frame) for the modular sheaf ! must already be chosen in Serre’s setting, this is a natural bit of data to consider. For the nearly overconvergent case, we take preimages of the Andreatta– Iovita–Pilloni torsors for overconvergent Hilbert modular forms in the space of frames for the de Rham cohomology. This construction works for elliptic modular forms, and seems eminently generalizable to be used for the automorphic forms on more groups, such as Hilbert–Siegel and hermitian modular forms. The author of this dissertation is currently working on a sequel extending his approach to the higher rank setting, in particular Hilbert–Siegel modular forms. 5 1.4 Notation and Conventions Fix a totally real field F of degree [F : Q] = d > 1. Its ring of integers will be denoted OF , and its inverse di↵erent will be denoted d1. Its Galois closure is Gal F , with ring of integers OGal. For a p-adic local field K, let OK be its ring of integers, and m for the maximal ideal in OK . We also write O1 = OK/m for the residue field, and Om = OK/mm for the local ring extending it. When the local field K contains GalF as a subfield, we say that K splits F . In this case, there are d independent embeddings of F into K. The set of such embeddings is denoted I. Given a prime ideal p of F lying over p in Q, write Fp for the completion of F in the p-adic metric. This is a finite extension of Qp. Write the local inverse di↵erent as d1p . For any integer N , let µN denote the group of Nth roots of unity. 6 CHAPTER II HILBERT MODULAR FORMS Fix an integer N . In this section, we give a geometric construction of Hilbert modular forms over F of level 1(N), and a comparison to the theory in terms of holomorphic functions. This serves as a base for our geometric construction of nearly Hilbert modular forms in Chapter III. 2.1 Weights Let K be a local field that splits dF , so that F ⌦Q K = K ; in this section we take K = R, but we state it in generality so that we may use this in Chapter IV. Let I denote the set of embeddings F ! K, noting that there are d distinct embeddings. Consider the algebraic group T = ResO /Z Gm. The weight space ofF classical Hilbert modular forms is the space of characters of the split torus T/K . Each embedding : F ! K determines a character of T/K via the formula : T ⇥/K(R)! R , (c⌦ r) = (c)r. All algebraic characters of T/K can be written in terms of these characters - for any character , there exists some tuple of integers (k) such that Y (c⌦ r) = (( kc)r) . X We often write such a character as a formal sum  = k. In this way, we 2I identify the space of characters of T/K with Z[I], the free abelian group on the set I of embeddings F ! K. 2.2 Moduli Setup Let c be a fractional ideal of F , p a prime number, and N 5 an integer prime to p. A c-polarized Hilbert–Blumenthal abelian variety (HBAV, we suppress c from the notation) defined over a base scheme S is a tuple (A, ◆, ,) where 7 – A! S is an semi-abelian scheme1 of relative dimension d, – ◆ : OF ! End(A) is a ring homomorphism known as a real multiplication on A, – : ⌦ d1 µN Z ! A is a closed embedding known as a level structure, and – : A⌦OF c! _A is a c-polarization. If c is principal, we say (A, ◆, ,) is principally polarized. Often, we will use A as a shorthand for the entire string. Let S be a scheme over SpecZ [ 1N ], X ! S be the moduli space of HBAVs over S, and ⇡ : A ! X be the universal HBAV over X. Remark 2.2.1. In the definition of HBAV, we really want A to be an abelian variety, rather than a semi-abelian scheme. If we take this alternate definition, and write Y as the moduli space of these varieties, then X is a projective toroidal compactification of Y . Write C = X \ Y , and refer to C as the boundary or the cusps. Allowing semi-abelian schemes in the definition of HBAV is similar to talking about the moduli space of generalized elliptic curves and the universal generalized elliptic curve when discussing elliptic modular forms. The Hodge bundle is ! = ⇡⇤⌦ 1 A ./X This is locally free of rank d as a sheaf of OX-modules, with an action of OF induced by the real multiplication. There is a largest open subscheme RX of X, such that ! restricts to a locally free sheaf of OF ⌦Z OXR-modules. If we use an ane scheme S = SpecO as our base and every prime number that ramifies in 1A group scheme which is an extension of an abelian variety by a torus 8 F is invertible in O, then RX = X. On the other hand, if O is a finite field of characteristic R Rp, and p ramifies in F , then X is of codimension 2. The space X is known as the Rapoport locus, and the condition that a HBAV A corresponds to a point of RX is called the Rapoport condition. A reference for this is [AIP16]. Recall that, in Section 2.1, we fixed a local field K that splits F . Over X/K , M the bundle ! decomposes as a direct sum ! = ! , so that each element c 2 O F 2I acts on ! as multiplication by (c) 2 K. 2.3 Hilbert Modular Forms For a tuple of positive integers  = (k) 2 Z1[I], we can form the line bundle over RX by /K ⌦k ⌦k ! := 1!1 ⌦ · · ·⌦ d !d . (2.1) Definition 2.3.1. A Hilbert modular form of level 1(N) and weight  = (k) is a global section of ! . The K-vector space of Hilbert modular forms of level 1(N) and weight  is thus 0H ( RX ,! ).  Write ⇥T := IsomXR,F (OF ⌦Z OXR ,!) for the RX -scheme whose points! are OF ⌦ OXR-linear isomorphisms from OF ⌦ OXR to !. We view this as the frame bundle of ! taking the OF -module structure into account. Over RX , ! is a vector bundle whose fiber over an open set U looks like OF ⌦OU . The points of ⇥T! correspond to bases for this free OF ⌦ OU -module: for any 2 ⇥↵ T , the element! ↵(1 ⌦ 1) gives a basis for ! as an OF ⌦ OXR-module. This space ⇥T has an action! of T ⇠/XR = AutXR,F (OF ⌦OXR) by precomposition: for any g 2 T and any ↵ 2 ⇥T ,! g · ↵ is the composition OF ⌦OXR ! g OF ⌦OXR ! ↵ !. 9 In fact, ⇥T is a T-torsor. This implies that its sheaf of functions of is graded by the ! characters of T/K . The graded portion corresponding to the weight  = (k) is the line bundle ! defined in Equation (2.1), and we find that the ring of Hilbert  L modular forms is the ring of functions on ⇥T , 2Z[ ] ! = O ⇥ .!  I  T! The grading is defined as follows. Each line bundle ! can be recovered from  O ⇥ by considering the homogeneous functions with the property that for any g 2 T! T ⇠, any HBAV A, and any frame ↵ : OF ⌦Z OXR ! !, ⇣Y ⌘ f(A,↵g) = ( )k g ( ) = ( 1f A,↵  g )f(A,↵). (2.2) Somewhat less explicitly, we may view these as functions from ⇥T to the rank one ! representation W of T, defined over K. The action of T is g · w = (g)w, and the functions are homogeneous in the sense that f( 1A,↵g) = g · f(A,↵). Remark 2.3.2. Homogeneous functions on ⇥T which are homogeneous as in ! Equation (2.2) are identified with sections of ! as follows. The pullback of ! from  R to ⇥X T is canonically trivialized by ↵, giving a trivialization ↵ of ! for each ,! N and thus a trivialization ↵ of ! with ⌦k↵ (1) = ↵ (1) 2 ! . Acting on the   trivialization by some g 2 T(K) sends ↵ to (g)↵, and (g↵) = (g)↵. Take a section v of ! defined on all of RX , and pull it back to a  /K section over ⇥T . Its value v(A,↵) at a point corresponding to a HBAV A with a ! trivialization ↵ of its cotangent bundle is a multiple f(A,↵) of the canonical basis ↵(1) for the line bundle ! . Since v was pulled back from a section defined over R X , we have that v(A,↵) = v(A,↵g) for any g 2 T(K). But the canonical trivialization of the fiber of ! over (A,↵g) is (g) times that of the canonical  trivialization over (A,↵). Thus we have f(A,↵g) = ( 1 g )f(A,↵). (2.3) 10 Thus we give the following alternate definition for Hilbert modular forms. Definition 2.3.3. Let R0 be a ring. A Hilbert modular form of level 1(N) and weight  = (k), defined over R0 is an algebraic function f 2 O ⇥ which satisfiesT! the homogeneity property of Equation (2.3). Write the space of such functions as O ⇥ []. T! Remark 2.3.4. While we defined Hilbert modular forms defined over a ring R0, we can patch this definition together over ane open sets to give a definition for Hilbert modular forms defined over some scheme S. Following Katz in [Kat78, Section 1.2], we give the following interpretation of Definition 2.3.3. For any R0-algebra R, and any HBAV A defined over R equipped with a trivialization ↵ of its cotangent bundle, we should get a number f(A,↵) 2 R subject to the following conditions: – the number f(A,↵) 2 R depends only on the isomorphism class of the pair (A,↵); – f commutes with extension of scalars, in the sense that, for any R0-algebra morphism i : 0R! R f(A⇥SpecR Spec 0 ⌦ 0R ,↵ R R ) = i(f(A,↵)); – and f satisfies the homogeneity condition from Equation (2.3), ( ) = ( 1f A,↵g  g )f(A,↵). Many important examples of Hilbert modular forms, such as Eisenstein ⇥ ⇤ series, are in fact defined over some finite extension of Z 1 . However, the analytic N picture involving holomorphic functions on a symmetric space in fact recovers 11 Hilbert modular forms defined over C. One should keep this in mind when reading the next section, in which we give this perspective. 2.4 Holomorphic Hilbert Modular Forms In this section we connect the geometric construction above to the view of Hilbert modular forms as holomorphic functions on a symmetric space. Following [Kat78], let OGal denote the ring of integers in the Galois closure of F . We have d distinct embeddings : O ! OGalF ; For any OGal-algebra A, we get a ring homomorphism Y OF ⌦Z A! A, n⌦ a 7! ((n)a). When is a flat OGalA -module, this is an injection. When the discriminant of F is invertible in A, it is a surjection. In particular, when A = R is the real numbers, Q we get a ring isomorphism OF ⌦Z R ! R. Note that R is an OGal-algebra because F is totally real. Writing R as an OGal-algebra is the same as fixing an embedding OGal ! R, which then identifies the embeddings OF ! OGal with the real embeddings of F . Thus we may index the product over the set I of real embeddings of F . Let h = {z 2 C | Im(z) > 0} be the complex upper half-plane, with its usual action of the group GL+2 (R) of invertible 2 ⇥ 2 matrices with real coecients and positive determinant given by the formula 0 1 · az + b B a bC z = , for all = @ A 2 GL+2 (R). cz + d c d 12 Y Then write hF = h for the product of d copies of h indexed by the real 2I embeddings of F ; it has a similar action of GL+2 (OF ⌦ R) given by the formula0 1 ✓ ◆ (a)z + (b) Ba b· C ( z) = , for all = @ A 2 GL+2 (OF ⌦ R). (c)z + (d) c d Fix some N 5, and let 1(N) be the following congruence subgroup. 8 9 >>>0 1> a, d 2 1 +NO >> < F =>>Ba bC 1(N) = >@ A 2 SL2(F ) 1 b 2 c d .> :> > > c d >> c 2 >Ncd1 ; Remark 2.4.1. In the case of modular forms (F = Q), congruence subgroups must be subgroups of SL2 Z. Note however that 1(N) need not be a subgroup of SL2 OF . Rather, we should allow congruence subgroups to be commensurable with that group: there is a common subgroup H of both and SL2 OF which has finite index in both. For the subgroup 1(N) above, let b be an integral ideal of OF contained in both c1d and cd1, and use 8 9 >>>0 1> > a, d 2 1 +NO > > H = >@ A 2 SL2(F ) b 2 b .> >>> >: c d 2 b ; >> c N The reader may check that this is indeed a a common subgroup of SL2 OF and 1(N), and that it has finite index in both. To each z = (z) 2 hF and each fractional ideal c, we associate a complex c-polarized HBAV (Az, ◆z, z,z) along with a OF ⌦C-basis for its Lie algebra. This will give us a map hF ! ⇥T (C), so that we can pull back complex modular forms! to be functions on hF . For a fractional ideal c and z 2 hF , let Lz = d 1 + c1z ⇢ OF ⌦ C. 13 This lattice can be equipped with a c-polarization associated to the alternating pairing h 1, i : Lz ⇥ Lz ! d c1, ha+ bz, c+ dzi = ad bc. This descends to a c-polarization z on the complex torus Az = (OF ⌦ C)/2⇡iLz. This torus has a natural OF -module structure given by acting on the first factor. We can also give a natural level structure. Identify the N -torsion of Az with the quotient 1 1 1 1Lz/Lz. This has a subgroup d /d . Then we can use the levelN N structure ✓ ◆ 1 µN ⌦ d1 ! Z/Z ⌦ 1 1 1 1 ! 1d = d /d , Lz/Lz = A[N ]. N N N The first map is the inverse of the exponential isomorphism k + Z 7! 2⇡ik/Ne . N Notice that by writing Az = (OF ⌦ C)/2⇡iLz, we are identifying Lie(Az) = OF ⌦ C. Thus, in order to trivialize the cotangent bundle, we just have to give a functional OF ⌦ C ! OF ⌦ C. We choose the identity map, and denote it by dw. We think of it as the di↵erential that sends each path in OF ⌦ C to its endpoint minus its starting point. Let ↵ : OF ⌦ C! ⌦1 be the trivialization sending 1⌦ 1 toA dw. This gives an embedding : h ⇥F ! T (C). Since we have defined (complex)! Hilbert modular forms as functions on ⇥T (C) in Definition 2.3.3, we can pull ! back complex Hilbert modular forms to be functions on hF . We give the following classical definition of holomorphic Hilbert modular forms, and then prove that all such forms are the pullback of Hilbert modular forms defined over C in the sense of Definition 2.3.3. 14 Definition 2.4.2. A holomorphic Hilbert modular form of level 1(N) and weight  = (k) is a holomorphic function hF ! C satisfying the homogeneity condition ! 0 1 Y Ba b· k Cf( z) = ((c)z + (d)) f(z), for all = @ A 2 1(N). c d If F = Q, then we also need a growth condition at the cusps. Write M(N) for the vector space of holomorphic Hilbert modular forms of level 1(N) and weight , and recall from Definition 2.3.3 that the set of Hilbert modular forms of level 1(N) and weight  can be written O ⇥ []. LetT! ⇤ : O ⇥ [] ! M(N) be the pullback along the map : h ! ⇥T (C) describedT! ! above. ⇠ Theorem 2.4.3. The pullback ⇤ gives an isomorphism O ⇥ [] !M(N).T! Proof. First we show that the image of ⇤ actually lands in M(N). Fix f 2 O ⇥ []. Certainly ⇤ f is a function hF ! C; we omit the proof that it isT! holomorphic. Rather, we should focus on the homogeneity property from Definition 2.4.2. For simplicity, we begin with the assumption that f has level 1(1). Pick some arbitrary 2 1(1),0 1 Ba b@ C= A 2 O 2 cd1 2 c1 , a, d F , b , c d. c d ⇣ ⌘ In particular, · = (a)z+(d) z ( ) + ( ) . Recall that elements of OF are included intoc z d ⇠ Q OF ⌦ C = C by n !7 ((n)). Since addition and multiplication happens componentwise, and each is a ring homomorphism, we may even write · z = ⇣ ⌘ az+b = (a)z+(d) . cz+d (c)z+(d) 15 Note that 2 1(1) gives a linear isomorphism from OF ⌦ C to itself that sends Lz into itself; thus Lz = zc 1 + d1 = = ( + )c1 + ( + )d1Lz az b cz d . ⇣ ⌘ Since Lz = ( 1 1 1 az+b 1 1 az+ b)c +(cz+d)d , we have that (cz+d) Lz = + c +d =cz d L·z. This gives us a homothety of lattices between Lz and L·z, which descends to an isomorphism A ⇠z = A·z. However, the rescaling by cz + d means that the pairs (Az, dw) and (A·z, dw) are not isomorphic. In fact, we get an isomorphism between the pairs (Az, dw) ⇠= (A·z, (cz + d)dw). Using the homogeneity property, we find f(Az, dw) = f( 1 A·z, (cz + d)dw) =  (cz + d)f(A·z, dw). The first term in the equality is ⇤ (f)(z), and the last is 1 ( ⇤cz + d) (f)( · z). Unfolding the definition of the character , we get the desired transformation property ! 0 1 Y Ba bC f( · z) = (( kc)z + (d)) f(z), for all = @ A 2 1(1). c d Thus, if f is homogeneous in the sense of Definition 2.3.3, then ⇤ f is homogeneous in the sense of Definition 2.4.2, so ⇤ f 2M(N). Now, we need to prove that ⇤ is a bijection, which we do by giving an inverse map. Every element of O ⇥ [] is determined by its values on the fibers T! over the open locus Y (C) ⇢ X(C) without the cusps. Now we just have to see that, given the function ⇤ f , we can recover f ; i.e., given the values f(Az, dw), we can find f(A,!) for any complex abelian variety A and any nonvanishing di↵erential ! on A. Note that the integral homology H1(A,Z) has an action of OF induced by the action of OF on A. A choice of di↵erential ! on A gives us the following lattice 16 L! ⇢ OF ⌦ C: ⇢Z L! = ! | ⌘ 2 H1(A,Z) . ⌘ Then A is isomorphic to A! = (OF ⌦ C)/2⇡iL!, and in fact, the pair (A,!) is isomorphic to the pair (A!, dw). One can choose an embedding d1 ! L! such that d1L!/ is torsion-free. So this quotient is projective, and thus is isomorphic to c1 by the classification of projective OF -modules. We can split this projection to write L! as a direct sum of two submodules; the first isomorphic to d1, and the second isomorphic to c1. Write them as (1)z d1 and (2)z c1. We can tweak (2)z so that (2) (1) z /z has positive imaginary part by replacing (2) z by its negative if necessary. (1) Then = z 2 h (2)z (2) F , and z Lz = L!.z This gives us a homothety of lattices between Lz and L!, which descends to an isomorphism A ⇠z = A. Taking the di↵erentials into account, we find that the pairs ( (2)Az, z dw) and (A,!) are isomorphic. Thus we have f(A,!) = ( (2))1 z f( (2) 1 ⇤Az, dw) = (z ) (f)(z), and f is completely determined by ⇤ (f). Thus ⇤ is invertible. For the case when f has level 1(N), a similar proof holds. We just have to see that 1(N) is exactly the set of elements g 2 G(R) such that the action of g preserves both the isomorphism class of the abelian varieties Az and the level structure. The level structure is the embedding : 1 d1/d1 ! 1 Lz/Lz; thisN N should be thought of as taking the OF -module isomorphism 1 Lz/Lz ! Az[N ] andN restricting it to 1 d1 d1 ⇢ 1/ Lz/Lz. Any 2 1(1) gives an automorphismN N of 1 Lz/Lz, and · is the level structure given by precomposing by thisN automorphism. Note that 2 1(N) if and only if0 1 Ba b@ C = A 1 1, a, d 2 1 +NOF , b 2 cd , c 2 Nc d. c d 17 Thus, for any n 2 d1 and 2 c1m , · should send n + m + to an+cmz L + N N z N bn+dm z + Lz. But an+cm = n + (a1)n+cm 2 n + d1, since (a 1)n and cm are N N N N N both elements of d1N . This is enough to see that preserves the level structure, since only cares about the restriction of this isomorphism to elements of the form n + d1 for 2 d1n . N Remark 2.4.4. Essentially, the proof that ⇤ is bijective relies on two facts. First, the image of the composition h ! ⇥F T ! X(C) is Y (C), and f is determined! by its values on the fibers over Y (C). Second, once you know the value of f on one element of each fiber, you can find the values over the entire fiber by using the homogeneity condition. 18 CHAPTER III NEARLY HILBERT MODULAR FORMS In Chapter II, we explained how to view Hilbert modular forms as geometric objects. Here, in we extend that construction to nearly Hilbert modular forms. In Section 3.1 and we discuss an extension of the modular sheaf that is important for this construction. In Sections 3.3 and 3.4, we build the sca↵olding a di↵erential operator r for each real embedding of the field F , using the Gauss–Manin connection as a key ingredient. Finally, in Section 3.6, we shift our focus to holomorphic functions and the classical Maass–Shimura operators . The section  ends by setting up a “realization functor” ⇤ that assigns a nearly holomorphic function to each nearly Hilbert modular form as constructed in Section 3.5. Finally, we prove that our operators r and correspond in the sense that the following square commutes. ⇤ ⌫ ⌫ N N   r  ⇤ ⌫+ N ⌫+N +2 +2 Here ⌫N is the space of nearly Hilbert modular forms of weight  and type ⌫, as  constructed in Section 3.5; and N ⌫ is the space of nearly holomorphic Hilbert  modular forms of weight  and type ⌫, as constructed in Section 3.6. This serves as a motivation for the construction of r in Section 4.6, since that will be entirely analogous to the construction of r here. For an application of to the values of p-adic L-functions for CM fields using Damerell’s formula, see  the Appendix A. 19 3.1 The de Rham Sheaf Another important sheaf is the relative de Rham cohomology 1H (A/X) = dR 1 ⌦•R ⇡⇤ A of A ! X. This sheaf has a natural subsheaf H which is locally free of/X rank 2 as a sheaf of OF ⌦Z OXR-modules;1 over K, it fits into an exact sequence of sheaves of OF ⌦Z OXR-modules known as the Hodge filtration: 0! ! ! H! _! ! 0. (3.1) Over K, ! and its dual _! are locally free of rank 1 as sheaves of OF ⌦Z OXR- modules, so they are projective. Thus the sequence splits, though non-canonically. In addition, H admits a nondegenerate alternating pairing with respect to which ! is a maximal totally isotropic subspace, which gives the explicit isomorphism between H _/! and ! . L Finally, we note that H splits as a direct sum H = H as ! did. This gives a similar exact sequence for each embedding : 0! ! ! H ! _! ! 0. This sequence gives a filtration ! ⇢ H of H, which induces filtrations on Symk H for each r k kr rk, with Fil Sym H = ! ⌦ Sym H. We also get a filtration N on SymH := Symk H indexed by the partially ordered set Z0[I], where a N pure tensor ⌫  r ks is in Fil Sym H for ⌫ = (r) if s 2 Fil Sym H for all . In particular, Fil0 SymH = ! , where ⌫ = 0 means ⌫ = (0). 3.2 Associated Bundles So far we have discussed two important bundles ! and H on X. In addition, we used ! to build the bundles ! , whose sections are Hilbert modular forms of  weight ; we explicitly related these bundles to spaces of functions on the frame 1In fact, 1HdR(A/X) agrees with H over RY . This H is a canonical extension of 1 RHdR(A/Y ), a locally free sheaf of OF ⌦Z OF -modules of rank 2, to such a sheaf defined over all of RX . 20 bundle ⇥T . In this section, we will give a more general version of this construction ! of so called “associated bundles” which will be built out of the whole H. We start with three groups. Write T = AutF,XR(!) for the automorphism group of ! as an OF ⌦ OXR-module. Over an open set where ! is trivial, T is isomorphic to ResO /Z Gm. Then let G = AutF,XR(H) be the automorphismF group of H as an OF ⌦ OXR-module. Over an open set where H is trivial, G is isomorphic to ResO /Z GL2; this explains its name, since we have been calling thatF group G since Section 1. Finally, write Q = AutF,XR(! ⇢ H) for the subgroup of G = AutF (H) consisting of automorphisms fixing the sub-bundle !. If we restrict to an open set where both ! and H are trivial, we can choose a basis for H whose first element is a basis for !; in this basis, the group G is simply a matrix group, and Q consists of the upper triangular matrices. For each of the three groups, we have an associated torsor. As before, let ⇥ ⇥ T = IsomX,F (OF ⌦ OXR ,!), noting that T is a T-torsor. We also consider the! ! G-torsor ⇥TH = Isom 2 X,F ((OF ⌦OX) ,H) and the Q-torsor ⇥,+TH = Isom+ ((OX,F F ⌦ O 2X) ,H), where the + superscript denotes the fact that we should only consider isomorphisms that respect the filtrations; i.e., isomorphisms ↵ : (OF ⌦ OX)2 ! H so that the restriction ↵|OF⌦OX to the first component is an isomorphism OF ⌦ OX ! !. If we think of these isomorphisms as simply giving bases for the fibers of H, we can think of ⇥,+TH as the subobject of ⇥TH consisting of bases whose first vector is a basis for ⇥!, and T as the quotient of ⇥,+TH obtained by forgetting the! second basis vector. These maps of torsors correspond to the maps of groups: the first to the inclusion Q ! G, and the second to the projection Q ! T which picks out the top left entry of the upper triangular matrix. 21 Finally, we explore the concept of “associated bundles.” Let ⇥T ! S be a torsor for some group H, and let V be a representation of H. Then we say that the contracted product ⇥T ⇥H V is an associated bundle, where sections of the contracted product over an ane open U = SpecR are homogeneous functions ( ⇥ 1T ⇥H V )(U) = {v : OT⇥(U)! V (R) | v(↵h) = h · v(↵) for all h 2 H(R)}. Example 3.2.1 (Associated Bundles to !). In Remark 2.3.2, we showed that sections of the line bundles ! are certain homogeneous functions on ⇥T .  ! Specifically, write  = (k). This corresponds to a one-dimensional representation Q ( k⇢ ,W) of T given by the formula ⇢(g) = (g) . Then ! is given as a  contracted product by = ⇥! T ⇥T W ! . These bundles can in fact be recovered from ⇥,+TH as well. Let W also denote the inflation to via the map ! T. Then ⇥,+Q Q T ⇥ W ⇠ ⇥ ⇠H Q  = T ⇥TW = ! .!  3.3 The Gauss–Manin Connection The relative de Rham cohomology admits a Gauss–Manin connection, which extends to a connection on H with logarithm poles on the boundary C. We denote it by r : H ! H ⌦ ⌦1 (log ). It induces a principal connection on ⇥C TH , and aX connection on any associated bundle ⇥TH ⇥G V for any representation V of G. We can describe it in terms of the action of the Lie algebra. Specifically, let U = SpecR ⇢ X be an ane open for which H|U is trivial, and we pick D a derivation R ! R, viewing it as an element of TX(U).2 Then the covariant derivative in the direction of D is a linear map r(D) : H ! H, which also commutes with the action of O . Any frame 2 ⇥F ↵ TH gives a natural basis for the 2The exterior derivative d : R ! ⌦1U gives rise to a directional derivative d(D) : R ! R for any D 2 TX(U). The directional derivative in the direction of D is a derivation on R, and every derivation arises this way. 22 fibers, and r(D) is given in this basis by some matrix X(D,↵) 2 M2(OF ⌦Z R) (if U meets the boundary, it will actually be in M2(OF ⌦ZFracR) with logarithm poles over C). This is g(R), where g is the Lie algebra of G. The map ↵ 7! X(D,↵) is G-equivariant for the Ad action of G on g, in the sense that X(D,↵g) = Ad( 1g )(X(D,↵)) for all g 2 G. This is the standard condition for r to induce a principal connection on ⇥TH . One can see this formula at the level of linear algebra by using g as a change of basis matrix, and noting that the Ad action of G on g is given by conjugation. Let V be a representation of G, so that it is also a representation of g. Then the associated bundle ⇥TH ⇥GV acquires a connection whose covariant derivative rV can be described as rV (D)(f)(↵) = Df(↵) +X(D,↵) · f(↵). (3.2) The action on pure tensors f ⌦ r 2 V ⌦ R is D(f ⌦ r) = f ⌦ D(r) and X(D,↵) · (f ⌦ r) = (X(D,↵) · f)⌦ r. Remark 3.3.1. One might think of functions to V ⌦ R as R-linear combinations of vectors in V . The Df(↵) term in Equation (3.2) di↵erentiates the function r 2 R using the exterior derivative, while the X(D,↵) · f(↵) term di↵erentiates the sections of V . This shows that it is a natural form for a connection that might be easier to describe in other ways. The Gauss–Manin connection is also used to define the Kodaira–Spencer map. Consider the following composition. ⌦ ! ⌦H Id⌦r 1! ! ! ! !⌦H⌦⌦ (logC)! !⌦ _! ⌦⌦1 (logC)! ⌦1 (logC). (3.3) X X X 23 This morphism is surjective, and over it identifies ⌦1K (logC) with the summand X L ⌦ L2 ⇠ ! of ⌦2! . Write this isomorphism as KS: ⌦1 (log ) ! ⌦2C ! . We will also X use the projection KS : ⌦1 (logC)! ⌦2! .X Remark 3.3.2. There are natural Hecke actions on the space of sections of ⌦1 (logC) and on ⌦2! . The projection KS is not equivariant with respect to theseX actions. While we do not consider Hecke actions in this dissertation, it is important enough to the general theory that this fact should be mentioned. 3.4 (g, Q)-modules In the previous section, we described how the Gauss–Manin connection r on H descends to a covariant derivative on the associated bundles ⇥TH ⇥G V for any representation V of G. However, this does not, for example, descend to a connection on the associated bundles for ⇥T or ⇥,+TH . Seeing as Example 3.2.1! realized Hilbert modular forms as sections of associated bundles to ⇥T , we need ! something more to be able define a covariant derivative acting on the space of Hilbert modular forms. As it turns out, the torsor ⇥,+TH will be an important intermediary. Representations of Q will be necessary in order to define the weights of nearly Hilbert modular forms, and a compatible action of the Lie algebra g will give us the action of r. How exactly these actions should be compatible is codified in the notion of a (g, Q)-module; to define them, we follow the exposition of [Liu19a]. Remark 3.4.1. The idea of a (g, K)-module has existed in the theory for much longer, especially in the representation theoretic approach to the subject. These are a fairly natural generalization; the action of the Lie algebra is necessary for a description of di↵erential operators, but rather than describing the weights in terms 24 of representations of the maximal compact subgroup K of the real points of G, we describe it in terms of the group Q. Let g = LieG and q = LieQ be the Lie algebras of G and of its Borel subgroup Q, respectively. Definition 3.4.2. A (g, Q)-module defined over an algebra E is an algebraic representation V of both g and Q on locally free E-modules, satisfying the following compatibility conditions. 1. The action of q ⇢ g is the same as that induced by Q. 2. For any g 2 Q, X 2 g, and v 2 V , Ad(g)(X) · v = g ·X · 1g · v. Any representation V of G has a natural action of g and of Q. These actions will be compatible in both senses above. In fact, any finite rank (g, Q)-module arises this way, c.f. [Liu19a, Remark 2.7]. In order to produce a (g, Q)-module, one can start with a representation V of Q and form a basis of g whose first elements are a basis of q ⇢ g. The elements of q act in the way induced by Q, so one only needs to give formulas for the action of the rest in a compatible way. Remark 3.4.3. Another way to describe a (g, Q)-module V is to pick some ring R and extend the Q(R)-module V (R) to a representation Vb (R) of an open subgroup Qb(R) ⇢ G(R) containing Q(R). Since Qb(R) is open in G(R), its Lie algebra is g. One then checks that the action of g preserves V (R) as a subspace of Vb (R), and then calculates the formulas for the action, which will automatically be compatible. These formulas will only involve scalars from R, so they will define a (g, Q)-module 25 over R. If the scalars all come from a subring of R and the basis is defined over that subring, then the formulas will define a (g, Q)-module over that subring. This is done e.g. in [Liu19a, Section 2.3] (“a more conceptual proof” after Equation (2.5)) for R = Zp; we will use this method in Section 3.5 as well. Let V be a (g, Q)-module. The formula in Equation (3.2) shows that the Gauss–Manin connection on the bundle associated to a G-module is given in terms of the exterior derivative and the action of g. Thus one might hope that we can define a connection on V := ⇥,+TH ⇥Q V . This is most of the content of the following proposition. Proposition 3.4.4. Let V be a (g, Q)-module. Then there is an integrable connection rV on the associated bundle V := ⇥,+TH ⇥Q V given by the formula rV(D)(v)(↵) = Dv(↵) +X(D,↵) · v(↵). Proof. The formula certainly gives a map O ⇥ 1,+ ! O ⇥,+ ⌦ ⌦ (logC). In order TH TH X to show that it gives a connection on V , we have to show that the homogeneity property holds, rV(v)(↵g) = 1g · rV(v)(↵). In fact, it holds for each term separately, and we treat them separately. First, 1 Dv(↵g) = D(g · v(↵)) = 1g ·Dv(↵). The first equality is by the homogeneity of v, and the second by the chain rule for the exterior derivative d. Then, X(D,↵g) · v(↵g) = Ad( 1g )(X( 1 1 1D,↵)) · g · v(↵) = g ·X(D,↵) · g · g · v(↵). 26 This reduces to 1g · X(D,↵) · v(↵). Here we have used the second compatibility condition for (g, Q)-modules in the second equality. Thus we have rV(D)(v)(↵g) = Dv(↵g) +X(D,↵g) · v(↵g) = 1g ·Dv(↵) + 1g ·X(D,↵) · v(↵) = 1g ·rV(v)(↵). This is what we wanted to show. The connection is integrable because d and r both are. 3.5 Nearly Hilbert Modular Forms Nearly Hilbert modular forms are sections of certain vector bundles. In particular, there should be finite rank representations ⌫V of Q for all  2 Z[I] and  ⌫ = (r) 2 Z0[I] so that a nearly holomorphic Hilbert modular form of weight  and type ⌫ is a section of V⌫ = ⇥,+TH ⇥ ⌫Q V . Each individual ⌫V is not a (g, Q)-   module, which complicates the definition of the di↵erential operators. However, there is an inclusion ⌫ ⌫V 1 ⇢ V 2 whenever ⌫i = ( (i)r ) and (2) (1)r r for all , and  S the union = ⌫V V is a (g, Q)-module. It will satisfy g ⌫ ⇢ ⌫+V V . Thus, by⌫    Proposition 3.4.4 and the forthcoming Remark 3.5.4, we will obtain a connection S rV on V = V⌫ satisfying⌫  M r ⌫V(V ) ⇢ V⌫+ ⌦ ⌦2! ⇢ V 1 ⌦ ⌦ (logC)  X The specifics are laid out in the rest of the section. For now, we fix an auxiliary prime ` which is unramified in F/Q, L an `-adic field which splits F , and OL its ring of integers. Let I denote the set of embeddings of F into L. P We fix a weight  = k 2 Z[I], with W the corresponding 1- dimensional representation of T defined over OL. We build the (g, Q)-module V following the course laid out in Remark 3.4.3 and [Liu19a]. 27 We begin by giving names to important, non-algebraic subgroups of G(OL). We will use the neighborhood IG(OL) Q(OL), which contains the lower parabolic subgroup Q (OL) and the Levi subgroup H(OIG L),80 1 9 <> >Ba bC = IG(OL) = >@ A 2 G(OL) | c 2 pOF ⌦OL: > , c d ; 80 1 9 <> > Ba 0C = Q (OL) = >@ A 2 G(OL) | c 2 pOIG F ⌦OL: > , c d ; 80 1 9 >< >Ba 0O @ C = H( L) = > A 2 G(OL): > . 0 d ; Note that Q (OL) projects onto T(OL) by picking out the top left entry a. WeIG inflate the representation W(L) from T(OL) to Q (O ). We have a uniqueIG L choice of `-adic topology on the finite dimensional Banach space W(L); consider the `-adic analytic induction Vb(OL) = IndIG(OL) W(O ) (L), which is the set ofQI LG `-adic analytic functions : IG(OL) ! W(L) which are homogeneous with respect to the action of Q (OL) in the sense that (hx) = h · (x) for allIG x 2 IG(O L), h 2 Q (OIG L). It is a representation of IG(OL) by right translation: (g · )(x) = (xg) for all g, x 2 IG(OL). By the Iwahori decomposition, every coset (O ) 2 Q L x Q (OL)\IG(O )IG IG L can be written as 0 1 1 YO B CQ ( ) x = Q (O ) . I L I L @ AG G 0 1 This is for some unique choice of Y 2 OF ⌦ OL, so that each 2 Vb is determined by its values on these matrices. We note that the coset spaces Q (O )\I (O ) IG L G L 28 and H(OL)\Q(OL) are the same, down to the choice of representatives, and view Vb as an algebraic representation defined over OL which consists of `-adic analytic functions from the coset space H\Q to W . This allows us to write Vb (R) = W ⌦ Orig\ , where O rig \ is the space of `-adic analytic functions on the rigid space H\Q.H Q H Q For any OL-algebra R, we have natural coordinates on (H\Q)(R) ⇠= OF ⌦ R ⇠ Q = R called Y for each embedding : F ! .3L By the standard theory of rigid spaces (in e.g. [Nic07]), the ring of `-adic analytic functions on H\Q is Orig (SpR) = RhY i, the space of power series over R in the variables Y whose H\Q coecients go to 0 `-adically. One may explicitly compute the action of I (O ) on VbG L (OL), obtaining the formula 0 1 Ba b· C(g P )(Y ) = ( 1a+ Y c) · P ((a+ Y c) (b+ Y d)), g = @ A c d Here Y and the entries of g are viewed as elements of the ring OF ⌦ OL. The induced action of Q(OL) and g(OL) may be computed from this formula. For Q(OL), this is simple: substitute 0 for c in the formula above. This gives the induced action of q(O ) on VbL (OL). To describe the action of the rest of g(OL), Q we need a basis. Since g(OL) = gl2(OL), we may specify the a0ctions1of Q 0 0 { µ | ! } B C : F L , where µ is the element of gl2(OL) which is @ A in the 1 0 entry corresponding to , and the zero matrix in all other entries. With our fixed choice of coordinates Y, we find that µ · @P (Y ) = Y" · P ( 2Y ) Y P (Y ). @Y 3This is where we use the fact that our auxilliary prime ` is unramified. 29 Q Here " 2 Lie(T) = gl1(OL) is the tuple with a 1 in the entry corresponding to and 0 in all other entries, acting naturally on W. Specifically, " · w = kw for any w 2 W. Notice that the space of polynomials W[Y ] is preserved by the actions of both Q(OL) and g(OL), though it is not preserved by IG(OL). These are the algebraic functions on the scheme H\Q. We have thus defined a (g, Q)-module structure on V = W ⌦ OH\Q, where now OH\Q is the space of algebraic functions on the coset space H\Q. Over OL, this is the polynomial ring OH\Q(R) = OL[Y ] = OL[Y]. Now we fix an algebra R over the ring of integers in the Galois closure of F such that the discriminant of F is invertible in R. Over R, we may write V = W ⌦ OH\Q = W[Y ] where Y = (Y) is our natural set of coordinates, and we can extend any basis for q to a basis for g by using { µ }. We have actions given by formulas. For Q, 0 1 a b (g · B CP )( 1Y ) = a · P (a (b+ Y d)), g = @ A . 0 d For g, we specify the action of the µ . · @µ P (Y ) = Y" · P ( 2Y ) Y P (Y ). (3.4) @Y These formulas give compatible actions since they were compatible on Vb. Remark 3.5.1. The coordinates Y are simply indeterminates at this stage. Later, we will turn certain algebraic functions to V into smooth functions on hF by substituting Y with 1 s = 2 , where y is the imaginary part of our coordinatesiy z on hF . The notation can be confusing at first, since the same letter is used for a quantity and 12 times its reciprocoal, but it is standard as in e.g. [Liu19a].i 30 Remark 3.5.2. If we replace R by a subring O ⇢ R, then g(O) is a subalgebra of Q g(R) = gl2(R). Thus these formulas can still be used to describe the action, but we have to be careful about what elements of the direct product we consider. This is particularly important if p ramifies in F when R = K is our chosen p-adic field and O = OK is its ring of integers. Not only does the action of Q preserve the degree of the polynomial, it preserves the degree in each variable Y separately. Thus V has an exhaustive filtration indexed by the partially ordered set Z0[I], such that ⌫V is the space of polynomials of degree at most r in the variable Y. The Lie algebra q also L preserves the filtration. Write g = g over K. Since g is generated by q and µ , it preserves the degree as a function of Y⌧ for each ⌧ 6= , and raises the degree as a function of Y by at most 1 by the formula in Equation (3.4). We have Q g ⌫ ⇢ ⌫+V V . Unlike V, each piece ⌫V has finite rank (1+ r), meaning that its   associated bundle V⌫ = ⇥,+ ⌫TH ⇥Q V has finite rank as a vector bundle.  Definition 3.5.3. The sheaf of nearly holomorphic Hilbert modular forms of weight is V = ⇥,+  TH ⇥Q V. The sheaf of nearly holomorphic Hilbert modular forms of weight  and type is V⌫ = ⇥,+ ⌫⌫ TH ⇥Q V . The K-vector space of nearly  holomorphic Hilbert modular forms of weight and type is thus 0 ⌫ H ( ⌫X,V ).  Remark 3.5.4. Using the filtration, we can refine the statement of Proposition 3.4.4. Define the di↵erential operator r to be the composition V rV! V ⌦ ⌦1 1⌦K  (logC) S! V ⌦ ⌦2! ⇠X = V+2. Then r(D) is given by the action of X(D,↵) 2 g, where D is the projection of 2 ⌦1D (logC) onto the summand ⌦2! . Thus it sends V⌫ into V⌫+ ⌦ ⌦2 ⌫+! ⇠ V . X   = +2 31 Specifically, r raises the weight of a nearly holomorphic Hilbert modular form by 2 and its type by . Q Recall Remark 3.5.2, which stated that g(OK) ⇢ gl2(OK) may be a strict inclusion if K is a p-adic field and p ramifies in F . In this case, D may not be an integral element of the Lie algebra even if D is. So r may not actually act integrally. We discuss this in Section 4.7. Note that we have written r with no reference to  or ⌫, since they are P clear from context while is not. In addition, write  = k in order to make better sense of what character + 2 corresponds to. It is known (c.f. [Kat78, Lemma 2.1.14]) that r and r⌧ commute for any pair of embeddings and ⌧ . This is done by reducing to the case of working over C, where we have the explicit formulas which will be given in the next section. These formulas define commuting operators over C, which are transported to being commuting operators over any base ring. Thus we may unambiguously write r0 P for the di↵erential operator that raises weights by 0 = 2 0 k and types (k). 3.6 Nearly Holomorphic Hilbert Modular Forms In Section 2.4, we described holomorphic Hilbert modular forms over C in Y terms of holomorphic functions on the product hF = h of upper half spaces. We 2I then set up a realization functor by giving a map : h ⇥F ! T and pulling back.! In this section, we will do the same for nearly holomorphic Hilbert modular forms, with the added bonus that the di↵erential operator described in Remark 3.5.4 will correspond to the classical Maass–Shimura operator. The complex space hF has coordinates (z)2I ; for each 2 I, we get a pair of di↵erential operators @ and @ . One way to characterize holomorphic @z @z functions is to say that a smooth function f : hF ! C is holomorphic if and only 32 if @f = 0 for all . Here we generalize this to the notion of a nearly holomorphic @z function. r+1 Definition 3.6.1. A function f is nearly holomorphic of type ⌫ = (r) if @ f r+1 = @z 0 for each . A function f is nearly holomorphic if there exists a tuple of integers ⌫ = (r) such that f is nearly holomorphic of type ⌫. In particular, a holomorphic function is nearly holomorphic of type (0). Remark 3.6.2. There is another characterization of nearly holomorphic functions. Following the conventions in [Liu19a], write = 1s . Then a function f is nearlyz z holomorphic of type ⌫ = (r) if and only if it can be written as a polynomial in the variables s with degree at most r as a polynomial in s alone, and for which the coecients are holomorphic functions. There are no algebraic relations between these s’s, so this polynomial is uniquely determined by f . Remark 3.6.3. In [Liu19a], the author’s nearly holomorphic functions are polynomials in the variables sij, where sij is the ij entry of the matrix (z z)1. In contrast, in [Shi00, 13.2], the author uses ri to denote a set of functions that play this role, noting that we may take them to be the entries of this matrix. We note that in the Hilbert case, 1s = = 1 2 where y is the imaginary partz z iy of z. The theory is often presented in terms of this y; we have chosen to use s to follow [Liu19a] more closely. This y is not related to the coordinate Y that we chose for the coset space H\Q. In the vein of Definition 2.4.2, define nearly holomorphic Hilbert modular forms as follows. Definition 3.6.4. A nearly holomorphic Hilbert modular form of level 1(N), weight  = (k), and type (r) is a nearly holomorphic function hF ! C of type 33 (r) satisfying the homogeneity condition ! 0 1 Y · k B a bC f( z) = ((c)z + (d)) f(z), for all = @ A 2 1(N). c d If F = Q, then we also need a growth condition at the cusps. Write ⌫N (N) for the vector space of holomorphic Hilbert modular forms  of level 1(N), weight  and type ⌫. We also define the classical Maass–Shimura operator as in e.g. [Shi00]. Definition 3.6.5. The weight  Maass–Shimura operator at , denoted , acts on  functions by the formula 1 @ ( f)( ) = k kz s s f(z). (3.5)  2 ⇡i @z If f is a nearly holomorphic Hilbert modular form of weight  and type (r), then f is a nearly holomorphic modular form of weight  + 2 and type ⌫ + . Write  n for the composition  n = +2( 1) · · · .  n  This raises the weight by 2n and the type by n. Since the partial deriatives 0 commute, and @s = 0 when =6 ⌧ , we may unambiguously write  for the @z⌧  composition that raises the weight by 2 0 and the type by 0 for any 0 . Remark 3.6.6. The factor of 12 in Equation (3.5) is a normalization factor required⇡i to make this an algebraic operation. Our choice of which abelian variety Az is associated to each z 2 h, as well as the choice to use dw as the di↵erential, are other normalization factors; our choices here follow [Kat78]. With these choices, we compare to r directly, rather than requiring another normalization factor. Our comparison theorem will be built very similarly to Theorem 2.4.3. Specifically, we will construct a function h ! ⇥,+F TH such that the composition 34 h ! ⇥,+F TH ! ⇥T is the function we used in Section 2.4; we will call this lift as! well. Then pulling back along will give us a function valued in ⌫V ; setting Y = 0  will give us a W-valued function. Forgetting the action of T, this is simply a C- valued function. Write, somewhat abusively, ⇤0 for the composition of the pullback ⇤ with the projection ⌫V ! W that evaluates at Y = 0. Once we have established that ⇤0 is an isomorphism, we will show that the di↵erential operator r described in Remark 3.5.4 corresponds to the Maass– Shimura operator by showing that the following square commutes.  ⇤ ⌫ 0 ⌫ N N   r ⇤ ⌫+ 0 ⌫+ N +2 N +2 Following [Liu19a], we describe two bases for 1H (Az), and put themdR together to find a suitable basis that we may use to extend as previously defined. In this discussion we use the fact that 1H (A) is the C-linear dual of Lie(A) ⌦ dR R C, and that Lie(Az) ⇠= OF ⌦ C. We have a natural basis for Lie(A ) ⇠z = OF ⌦ C as an OF ⌦ R-module given by 1 and z. This extends to an OF ⌦ C-basis of Lie(Az) ⌦R C; let ↵ and be the elements of the dual basis, defined by ↵ (c1 + c2z) = c1, (c1 + c2z) = c2. We may also use the natural C-action on Lie(Az) to get a splitting of Lie(Az) ⌦R C = Lie(Az) Lie(Az). The space Lie(Az) ⌦R C has two actions of C: one on the first component and one on the second. Write Lie(Az) for the subspace on which these actions agree v⌦ c = cv⌦ 1; and Lie(Az) for the subspace on which they di↵er by complex conjugation, v ⌦ c = cv ⌦ 1. The dual space to Lie(Az) ⇠= OF ⌦ C is 35 simply the cotangent bundle ⌦1 ; we have a basis dw for this space, which is the Az simply the natural identification Lie(Az)! OF ⌦C. The complement of ⌦1 , whichAz we called Lie(Az), is also naturally identified with OF ⌦ C, and let dw denote the functional given by this identification. Neither of these pairs of functionals is quite the basis we want. In order for the chosen basis to be a lift of the function described in Section 2.4, we need to choose a basis with dw as the first element. Then, we need another element of 1 H (Az) to extend this to a basis of the whole space. We have two useful choices.dR First, we might choose a slight rescaling of the second basis, (dw,dw s). This rescaling is necessary for the basis to respect the pairing, and in fact this is the basis we choose to use when we define the extension : h ! ⇥,+ F TH . However, we might instead choose the basis (dw, ). This has the advantage that the di↵erential operator is given by an explicit element of the Lie algebra. Specifically, via the identification 1! ⌦O ! ⇠F = ⌦ , the basis dw gives a basis forX ⌦1 , and a dual basis D = (D) for TX . Writing dw and for the correspondingX basis of 1H (Az)() for each 2 I, we have that r(D)(dwdR ) = . Thus, in this basis, the di↵erential operator r acts via the Lie algebra element µ , for which we have an explicit formula. This also has the benefit of being a holomorphic basis: for a nearly Hilbert modular form f 2 O ⇥,+ ⇥Q V, we have that the functionTH z 7! f(Az, dw, ) can be written as a polynomial Pf (Y ) in the variables Y with holomorphic functions as coecients. The final piece of the puzzle, then, is to relate these two bases to each other. The action of Q gives us 0 1 0 1 B1 s 1 s (dw,dw s)@ AC B C= (dw, ) or, (dw,dw s) = (dw, )@ A . 0 1 0 1 36 In particular, if write f(Az, dw, ) = Pf (Y ), we have 0 0 11 B B1 sCC f(Az, dw,dw s) = f @Az, (dw, )@ AA 0 1 0 1 B1 s@ C= A · f(Az, dw, ) 0 1 0 1 B1 s = @ AC · Pf (Y ) 0 1 = Pf (s+ Y ) Thus ⇤0(f)(z) = Pf (s) is a polynomial in the variables s with holomorphic functions of z as coecients. If f has type ⌫ = (r) in the sense that f 2 O ⇥,+⇥T QH ⌫ ⇤ V , then 0(f) is nearly holomorphic of type ⌫ by our alternate characterization in Remark 3.6.2. This process is reversible since Pf (Y ) can be recovered from Pf (s). Thus, to prove that ⇤0 : V⌫ ! ⌫N (N) is a valid realization functor, we just have to show that  ⇤ 0(f) satisfies the transformation property, ⇤ ⇤ 0(f)( · z) = (cz + d)0(f)(z). We have (Az, dw,dw s) ⇠= ( 1A·z, (cz + d)dw,(cz + d) dw s). Similar to replacing dw by (cz + d)dw, the action of replaces dw by (cz + d)dw and s by s| + |2 , which has the net e↵ect of replacing the product dw s by (cz +cz d 1 d) dw s. Thus we get the required transformation property by using the formula 37 for the action of Q on V:0 1 B( + )1⇤ cz d 0 C 0(f)( · ) = @ A · ⇤ 1 ⇤ z 0(f)(z) = (cz + d) 0(f)(z). 0 (cz + d) We have proved, Theorem 3.6.7. The map ⇤ : V⌫0 ! ⌫N (N) is an isomorphism.  We are left to show that this intertwines the di↵erential operators r and .  Theorem 3.6.8. The operator r is the correct analog of the Maass–Shimura operator , in the sense that ⇤ ⇤  0 r =  0. Proof. We have formulas for each operator, and we compare them. Let Pf (Y ) be the polynomial with coecients which are holomorphic functions of z such that f(Az, dw, ) = Pf (Y ). By the formula in Equation (3.4.4), write r · @P( ) ( d ) = f f Az, w, (Y ) + µ Pf (Y ). @z Here the operator @ di↵erentiates the coecients of Pf . Using the formula for the@z action of µ , we get r @P( f 2 @ · f) (Az, dw, ) = (Y ) + Y"Pf (Y ) Y Pf (Y ). @z @Y The action of " is just multiplication by k. Applying the realization functor, we get ( ⇤0 r @P @P )( f 2 f f)(z) = (s) + ksPf (s) s (s). (3.6) @z @Y Now we should compare this to ( ⇤ 0)(f). We have that ⇤0(f) = P f (s). Note that @s⌧ = 0 if 6= ⌧ , and @z @s @ 1 1= = = 2s . (3.7) @z 2 @z z z (z z) 38 Using the definition of and the product rule, we get  ✓ ◆@ @ @ ( · ⇤( ))( ) = k k ( ) = k k 0 f z s s Pf s s s P (s) + (P (s)) . (3.8) f f @z @z @z Here, the derivative in @Pf (s) acts only on the coecients of Pf ; on the other hand,@z @ (Pf (s)) refers to the derivative of the whole nearly holomorphic function Pf (s).@z The product rule gives @ @P @P @s @P @P ( ( )) = f ( ) + f ( ) · = f 2 fPf s s s (s) s (s). @z @z @Y @z @z @Y On the other hand, using Equation (3.7), we have @ k = k1@s +1 @= k k ks ks ks , s s = k s . @z @z @z Plugging the previous two equations into Equation 3.8, we get ( · ⇤ @Pf 2 @Pf  0(f))(z) = ksPf (s) + (s) s (s). (3.9) @z @Y Comparing Equations (3.6) and (3.9), we conclude. 3.7 Splitting the Hodge Filtration This section is included to give some intuition for why certain objects might show up in this theory. In it, we attempt to explain why the (g, Q)-module V is a natural object to consider by giving an interpretation of its relationship with W. It is not necessary in order to follow the results and proofs of this dissertation. We have not yet talked about one of the most important features of the theory of nearly Hilbert modular forms: the choice of splitting of the Hodge filtration. Recall from Equation (3.1) that the bundles ! and H fit into the following exact sequence of sheaves of OF ⌦ OX-modules known as the Hodge filtration: 0! ! ! H! _! ! 0 39 We also recall that we defined the group schemes G, H, and Q in terms of these objects. We had G, the group of automorphisms of H that preserve the pairing; H the group of automorphisms of !; and Q, the group of automorphisms of H that preserved ! as a subsheaf. Pick a trivialization ↵0 : OF ⌦ OX ! !, and extend it to a trivialization ↵ of H such that ↵(1, 0) = ↵0(1) and h↵(1, 0),↵(0, 1)i = 1. This gives a splitting of the Hodge filtration. Specifically, 0 _↵(0, 1) projects to a basis ! 2 ! ; every element of _! can be written uniquely as a constant multiple of this element, 0c! . Then we define an inclusion _⌘ : ! ! H by sending 0c! to c↵(0, 1). This induces a decomposition of H as an internal direct sum H = Span{↵(1, 0)} Span{↵(0, 1)} = ! _⌘ (! ). In this basis, we can write G as a group of 2 ⇥ 2 matrices, with Q as the subgroup of upper triangular matrices. We also embed T0into Q as1a subgroup H Bh 0 C of diagonal matrices; write h 2 T as the diagonal matrix @ A. The group H 0 1h preserves the direct sum decomposition H = _! ! . On the other hand, while Q does preserve the subspace ! ⇢ H, it does not preserve the complement ⌘ ( _! ); in fact, Q acts transitively on the set of such complements, while H is the subset that preserves each decomposition. Thus the set of splittings of the Hodge filtration is a Q-set, and the stabilizer of each splitting is H. So any choice of a splitting gives an isomorphism between the set of splittings of the Hodge filtration and the coset space H\Q. Now recall that the representation V of Q is the set W ⌦ OH\Q. We should think of this as the space of functions from H\Q to W, or once a choice of a splitting of the Hodge filtration is chosen, perhaps as the space of functions from the set of splittings of the Hodge filtration to W. Given an element of 40 ⇥,+ TH , we can think of the trivialization of H as giving us a trivialization of ! along with a choice of splitting of the Hodge filtration. (It also gives a basis for the complement _⌘ (! ), but for this section we will ignore this fact.) We get the following philosophical result: a nearly Hilbert modular form is a gadget that takes in a splitting of the Hodge filtration and splits out a Hilbert modular form. Note that a nearly Hilbert modular form f is an honest Hilbert modular form if it doesn’t depend on the choice of splitting. On the algebraic side, this happens if and only if it descends to a function on the quotient ⇥ ⇥,+T of TH . On the! complex analytic side, this happens if and only if the nearly holomorphic function f is actually holomorphic. When F = Q, this conversation is more explicit. Working over C, we have a natural splitting of the Hodge filtration given by H = !!, where ! is the space of holomorphic di↵erentials on the universal elliptic curve A, and ! ⇠= _! is the space of anti-holomorphic di↵erentials. The coset representatives of H\Q are matrices of the form 0 1 @B 1 YCA , Y 2 Hom( _ ⌦2! ,!) ⇠= ! . 0 1 There are only a few sections of ⌦2! , and none of them are holomorphic. In fact, they are all multiples of the function 3 12E2 = E⇡y 2 + 2 s, where E is a⇡i 2 normalized Eisenstein series of weight 2. As a function on h, it satisfies the weight 2 modularity condition. Now let f(z) be a holomorphic modular form of weight k, so = ks 1 @fkf 2 f + 2 is a nearly holomorphic modular form of weight k + 2. Since⇡i ⇡i @z 1 ( + 12ks) = 2⇡ik12 kE2 2 f 12 E2f +ksf also satisfies the weight k+2 modularity condition,⇡i their di↵erence @f 2⇡ik@f = 12 E2f satisfies the weight k+ 2 modularity condition as@z well, and it is holomorphic. This @f is the holomorphic projection of kf , and it is 41 an honest holomorphic modular form of weight k + 2 when viewed as a function on h. In the sense above, the nearly holomorphic function kf corresponds to a certain nearly modular form rf . This nearly modular form takes a splitting ⌘ of the Hodge filtration and spits out a modular form, which corresponds to a function on h. There is one splitting ⌘0 of the Hodge filtration for which the function on h corresponding to (rf)(⌘0) is actually the nearly holomorphic function kf , and one splitting ⌘holo of the Hodge filtration for which the function on h corresponding to (rf)(⌘holo) is actually holomorphic. All other splittings fall somewhere in between. 42 CHAPTER IV P -ADIC THEORY Let K be a p-adic field that splits F , as in Section 2.1. As established in Section 1.4, OK is its ring of integers, and m the unique maximal ideal in OK . For any m 1, let Om = OK/mm, so that in particular O1 is the residue field OK/m, which has characteristic p. In this section we work over the bases S = SpecOK and Sm = SpecOm. 4.1 Weights Recall the discussion of weights in the Archimedean theory, from Section 2.1. In the Archimedean theory, weights were characters of the algebraic group T = ResO /Z Gm. For the p-adic theory, weights are instead characters of the constantF group scheme T(Zp). The weight space W , a rigid analytic space defined over K associated to the algebra ZpJT(Zp)K. The Cp-points of W parametrize continuous homomorphisms T(Z ⇥p) ! C , which are the p-adic weights of Hilbert modular forms. As a rigidp analytic space, it is isomorphic to a finite disjoint union of open unit balls of dimension d, where each component is labeled by a (finite order) character of the torsion subgroup of T(Zp). A character of T(Zp) is called algebraic if it is in Z[I], and locally algebraic if it is the product of a finite order character and an algebraic character. Locally algebraic characters are dense in the weight space. For any value1 w 2 v(OF ), the rigid analytic group T has two important subgroups. The first, T0 , consists of units congruent to 1 modulo (p)w. The second, w Tw, is generated by T0 and T(Zw p). 1If p ramifies with index e, w 2 1eZ. Writing ( ) w p should make sense as an ideal of OF , and we will sometimes write wp abusively even if there is no such element. 43 Let A be an anoid algebra, and let ⇥ : T(Zp) ! A be a family of characters parametrized by the anoid space U = SpA. As asserted in [AIP16], such a character is w-analytic for some w, meaning that it factors as a composition as follows for some linear map : T0 Z logw F w exp( ⇥p) = 1 + p (OF ⌦ Zp) ! p OF ⌦ Zp ! pA ! 1 + pA ⇢ A . (4.1)w There is a universal character un : W ⇥ T(Z ⇥p) ! C , where un (x, t)p evaluates the character associated to the point x at the input t 2 T(Zp), corresponding to the natural character T(Zp) ! OKJT(Zp)K sending g to g. The universal character is 1-analytic. For a family of p-adic, or overconvergent, Hilbert modular forms, its weight should be a family of characters; if the family of p-adic Hilbert modular forms is parametrized by a rigid space U , so should be the family of weights. Any family of characters parametrized by a space U is specified by pulling back the universal character by a map U !W . 4.2 Canonical Subgroups for Ordinary HBAVs In this section we discuss ordinary HBAVs and their properties. Consider an HBAV A1 defined over the residue field O1, and a prime ideal p of F lying over p. Consider the pn-torsion of A1, [pnA1 ] = {a 2 A1 | x · a = 0 for all x 2 pn}. For our fixed A1, there are two possibilities for the sizes of these subgroups. Either # [pn] = Nm(p)nA1 for all n, or # nA1[p ] = 1 for all n. In the first case, we say that A1 is ordinary, while in the second we say that A1 is supersingular. 44 For an HBAV A defined over OK , we say that A is ordinary or supersingular depending on its base change A1 = A⇥OK O1 to the residue field. If A1 is ordinary, we say A is ordinary, and if A1 is supersingular, we say that A is supersingular. There is a test for whether or not A1 is ordinary depending on the value of a modular form, known as the Hasse invariant. Let q = Nm(p), Fr be the relative q-power Frobenius map (q)A1 ! A1 , defined by the following diagram: Frabs A1 Fr (q) A1 A1 Fr O absSpec 1 SpecO1 In this diagram, both maps A1 ! SpecO1 are the structure maps. The maps SpecO1 ! SpecO1 and A1 ! A1 are both the absolute Frobenius maps, which are the identity on points, but pulling back functions raises them to the qth power. The bottom right square is a pullback diagram which defines (q)A1 , and the relative Frobenius Fr fits in as the dashed arrow by the universal property of the pullback. Remark 4.2.1. In fact, (q)A ⇠1 = A1, which is easiest to see in terms of equations. Since A1 is an abelian variety, it is projective. Embed it into some projective space; it is the zero set of some collection of homogeneous polynomials {fi}i2I , and since it is defined over O1, the coecients of these polynomials are elements of O1. Then (q)A1 is the zero set of the polynomials { (q) f }i2I , where (q)f is thei i polynomial obtained from fi by raising all of its coecients to the q power. Since the coecients are in O1, raising them to the q power is the identity, and (q)A1 is the same subvariety of the same projective space as A1 was. This is one advantage of 45 using the relative q-power Frobenius map instead of p-power. For elliptic curves, this is spelled out in e.g. [Sil09, Example II.4.6]. We will not distinguish between the two for the rest of this section. The dual isogeny to the relative Frobenius, V : Â1 ! Â1 is known as verschiebung. Vershiebung is either separable, if A1 is ordinary, or inseparable, if A1 is supersingular at p for some p lying over p. Fix an HBAV A1 defined over an O1-algebra R1 such that ⌦1 is a freeA1 OF ⌦ R1-module of rank 1 (i.e., A1 satisfies the Rapoport condition). Then choose a trivialiation ↵ : OF ⌦ 1R1 ! ⌦ , and let ! = ↵(1). Using the decompositionA1 L ⌦1 ⇠= ⌦1 (), we get an R1-basis { 1!} for ⌦ , where ! is the image of !A1 A1 A1 after projecting to ⌦1 (). A1 The polarization induces a perfect, OF -linear pairing ⌦1 ⇥ T ! R1. LetA1 Â1 {⌘} be the dual basis to {!}, and notice that pullback by Fr gives a skew-linear map T ! T , such that Â1 Â1 Fr(c⌘) = qc Fr(⌘). (4.2) Notice that, since ( q c ) = ( )qc , this map is OF -linear. Thus, if ⌘ is dual to an element of ⌦1 (), so is Fr(⌘). Define the function Ha to be the coecientA1 Fr(⌘) 7! Ha(A1,↵)⌘. We note that this satisfies a homogeneity property: for any c 2 OF ⌦ R1, c · ↵ gives the new dual basis {(c)1⌘}. Using Equation (4.2), we see that Fr( ( )1 q c ⌘) = (c) Fr(⌘). Writing each side of the equation above in terms of the Hasse invariant, we get Ha 1 q(A1, c↵)(c) ⌘ = (c) Ha(A1,↵)⌘. 46 Multiplying both sides by (c) and comparing coecients, we find Ha(A1, c↵) = (c) 1q Ha(A1,↵). (4.3) Definition 4.2.2. For any abelian variety A1 as above and any trivialization ↵ of its cotangent bundle, define the Hasse invariant at , Ha, to be coecient Fr(⌘) 7! Ha(A1,↵)⌘ By Equation (4.3), we see that Ha is a Hilbert modular form of weight (q 1). For every prime p lying over p, let Ip be the set of embeddings F ! K which “belong to p.” Specifically, Ip = { 2 I | OF ! OK ! O1 factors through the reduction OF ! OF/p}. Then define the Hasse invariant at p, Hap, to be the product Y Hap(A1,↵) = Ha(A1,↵). 2Ip P These are Hilbert modular forms of weight 2 (q 1); we also call them partial Ip Hasse invariants. Finally, define the full Hasse invariant Ha to be the product Y Y Ha(A1,↵) = Ha(A1,↵) = Hap(A1,↵). 2I p|p This is a Hilbert modular form of parallel weight q 1. Remark 4.2.3. The partial Hasse invariants Ha are only defined over the Rapoport locus. However, as noted in [AIP16, Remark 3.1, page 11], the products Hap extend to be defined over the whole modular curve. An abelian variety A1 over O1 will be ordinary at p, i.e., V will be separable, if and only if Ha( 1A1,↵) 6= 0 for some trivialization ↵ of ⌦ . The homogeneityA1 property shows that one may check this condition for any choice of ↵. We say A1 is ordinary at p if Hap(A1,↵) 6= 0 for some trivialization ↵. The product formula 47 Q Ha = p Hap shows that A1 is ordinary at p if and only if it is ordinary at p for all p dividing p. Remark 4.2.4. When F = Q, this story simplifies somewhat, since an HBAV is just an elliptic curve. Here an elliptic curve is canonically polarized; let E be an elliptic curve, Fr : E ! E be the relative p-power Frobenius, and note that we can view its dual isogeny as a map V : E ! E. Since Fr has degree p, we can factor the multiplication by p map as [p] = V Fr, and since Fr is (totally) inseparable, it has trivial kernel, giving that the p-torsion is E[p] = ker[p] = kerV . This shows that #E[p] = #kerV , which is 1 if V is inseparable and p if V is separable. So we should find a way to determine whether V is separable or inseparable. In general, an isogeny : ! 0f E E is separable if and only if the pullback on cotangent bundles ⇤f : ⌦1 0 ! ⌦1 is nonzero, see e.g. [Sil09, III.4.2c]. Since theE E cotangent bundles are rank 1, we only have to check whether ⇤V is nonero, or that ⇤ V (!) 6= 0 for a single generator ! 2 ⌦1 . This is essentially what the Hasse E invariant does for us; more specifically, Ha checks whether or not the dual to this map is nonzero. Remark 4.2.5. Though we have taken the view that Hilbert modular forms should be thought of as functions on ⇥T , here it is useful to realize that the ! Hasse invariant is a section of a line bundle on X, the moduli space of HBAVs. Specifically, this view allows us to see that its zero locus, consisting of points that correspond to supersingular HBAVs, is a closed subscheme ssX ⇢ X. On the other hand, the complement of ssX , which consists of points that correspond to ordinary HBAVs, is a dense open subscheme ord ssX ⇢ X. We call X the supersingular locus, and ordX is the ordinary locus. 48 Let d1p be the local inverse di↵erent in the completion Fp. If A is ordinary at p, then as an algebraic group, we have A[pn] ⇠= d1 np ⌦ µpn ⇥OF/p . The maximal connected subgroup d1p ⌦ µpn is called the canonical subgroup of level pn of A, denoted Cpn = Cpn(A). For n1  n2, we have Cpn1 ⇢ Cpn2 , and the union over all n gives the canonical subgroup of level p1 of A, denoted Cp1 . For n  1, let Cpn be the maximal connected subgroup of [ nA p ]. This is the canonical subgroup of level n p , and it is precisely the subgroup of A[ np ] generated by all the groups Cpn for p dividing p. Remark 4.2.6. For all  1, the only point of d1n p ⌦ µpn over a field of characteristic p is the identity; this accounts for the fact that [pnA ] and OF/pn both have size Nm(p)n over the special point S1 of S, even though # nA[p ] = Nm( )2np at the generic point SpecK. In this sense, we can think of the canonical subgroup as the “kernel of the reduction map”. For any n < 1 and m < n, we have the following as in [CEF+16, Equation (9)]. It is valid for any ordinary abelian variety A. ⌦1 O = (Lie( D A)) = (Lie(A)[ n Dp ]) = (Lie( )[ nA p ])D = (Lie DC A/ m n (A)) Here A[ np ] is the connected component of the identity, which is the canonical subgroup Cn. We are allowed to consider the np -torsion because we are working over Om for m < n. Then, since the Tate module is isomorphic to the Lie algebra of a np -torsion group, (Lie ( DC A)) ⇠= ( DT C (A)) ⇠ Dn p n = C (A)⌦On m. This shows that the cotangent bundle ⌦1 O is isomorphic to the dual of theA/ m canonical subgroup DC (A) ⇠ n = O nF/p OF . This is also true in families; write 49 ord := ordX X ⇥OK Om, and Aord = Aord ⇥ O . Thenm m OK m ⌦1 ⇠ D ordAord O = C (A ). (4.4)m / m n m Remark 4.2.7. Let A ! T be a family of HBAVs parametrized by some S-scheme T . We view this as an HBAV defined over T . Then if A is ordinary, its canonical subgroup is also a family 0Cpn ! T . For any T ! T , we can define the fiber product A ⇥ 0T T which will also be ordinary; its canonical subgroup will be the fiber product 0Cpn ⇥T T . 4.3 p-adic Hilbert Modular Forms In Chapter II, we described Hilbert modular forms as functions on the torsor ⇥T , the frame bundle of the pushforward of the relative cotangent bundle ! 1 ! = ⇡⇤⌦A of the universal HBAV ⇡ : A ! X. In the p-adic setting, however,/X the cotangent bundle is not as well-behaved as one might hope. In Section 4.2 we described canonical subgroups and the Hodge–Tate map, which connected these canonical subgroups Cpn to the cotangent bundle ⌦1 . We will use canonicalA subgroups as a replacement for the cotangent bundle in the p-adic setting. Recall that the ordinary locus ordX is an open subscheme of X, and let Aord = A ⇥ ord be the universal ordinary HBAV. Since AordX X is ordinary, it has a canonical subgroup Cp1 , and the Hodge–Tate map gives an isomorphism ⌦1Aord ! Cp1 defined over ord X ⇥OK Om for any m. This gives rise to a morphism (not an isomorphism) ⌦1Aord ! Cp1 over OK . For all n < 1, let ⇥T n = Isom 1Xord(Cpn , d ⌦ µpn) be the set of OCp F -linear isomorphisms from the canonical subgroup Cpn of the universal ordinary HBAV Aord over the ordinary locus ord to d1X ⌦ µpn . This is a torsor for the group of O -linear automorphisms of d1F ⌦ µpn ; when n < 1, this automorphism group is isomorphic to AutF (d1 ⌦ µpn) ⇠= (O nO ⇥F/p F ) = T(Z n/p Z). 50 When n = 1, consider instead the formal scheme T⇥ ⇥C 1 = limT n . This isp Cp n a formal torsor for the group limT(Z n/p Z) ⇠ = T(Zp), known as the Igusa tower. A n function on T⇥C1 takes in a pair (A, u) of an HBAV A defined over an OK-algebra Rp and a trivialization 1u : Cpn ! d ⌦ µpn and returns an element of R. Definition 4.3.1. Let R0 be a complete and separated OK-algebra, and  a character of T(Zp). A p-adic Hilbert modular form of level 1(N) and weight  defined over R0 is an algebraic function f 2 OT⇥ which satisfies the homogeneityCp1 property ( tf A, g u) = ( 1g )f(A, u). (4.5) Write the space of such functions as OT⇥ [].Cp1 Remark 4.3.2. Let R1 be a normal OK-algebra which is p-adically complete, separated, and topologically of finite type. Let Rm = R1⌦OK Om. Given a classical Hilbert modular form f , i.e., a function on ⇥T , defined over R1, we now describe! how to view it as a p-adic Hilbert modular form. Each isomorphism : 1 ! d1u Cp ⌦ µp1 restricts to an isomorphism on the n p -torsion, un : Cp1 ! d1 ⌦ µp1 . We dualize, obtaining isomorphisms D 1u : (d ⌦n µpn)D ! DC n for each n. Now fix some m 1. When m < n, composing with thep isomorphism of Equation (4.4) we obtain, for any pair (A, u) of an ordinary HBAV A and a trivialization u : 1 1 DCp1(A) ! d ⌦ µp1 , a trivialization ↵n : (d ⌦ µpn) ! ⌦1 O . In fact, for a fixed m, this trivialization ↵n is independent of n (as long asA/ m m < n). This gives us an isomorphism ⇥ ⇠= ⇥ ⇥ ordT n T XR X . We also have thatCp ! (d1 ⌦ µpn)D ⇠= OF ⌦ Rm. Using this viewpoint, one sees that, in an appropriate category of locally ringed spaces, T⇥ 1 = ⇥ ord Cp limT ⇥XR X . (4.6)! m n 51 Thus, a p-adic Hilbert modular form f defined over R1 is a coherent system of classical Hilbert modular forms ( )1fm =1 defined over Rm with the caveat that wem only care about their values over the ordinary locus, rather than over the whole Rapoport locus RX . Notice that the homogeneity property in Equation (4.5) includes a transpose, as well as putting the group element on the opposite side of the trivialization compared to Equation (2.3). This is because of the step in the process above where we have to dualize the automorphism – the dual of tg u becomes Du g, which looks much closer to the form taken by the latter equation. In fact, this is how we check the homogeneity condition. Since T is a group of 1 ⇥ 1 matrices, the transpose map is just the identity; we could have left it out. However, in the case of higher rank groups, this is a crucial piece, so we include it here. The ring of p-adic Hilbert modular forms of level 1(N) is OT⇥ . By theCp1 discussion in Remark 4.3.2, the ring O ⇥ of classical Hilbert modular forms maps T! to the ring of p-adic Hilbert modular forms, simply by reducing the classical Hilbert modular form modulo mm for each m. This map is injective with dense image (see [Hid04]), which justifies saying that the ring of p-adic Hilbert modular forms is the completion of the ring of Hilbert modular forms; this is a good thing, as it was the initial motivation behind Serre’s introduction of p-adic modular forms in [Ser72]. 4.4 Overconvergent Loci and Partial Canonical Subgroups For each of the g primes p of F lying over p, we have a partial Hasse invariant Hap defined in Definition 4.2.2. This is a Hilbert modular form defined over the finite field O1. We would like to choose a lift of this to a Hilbert modular 52 form defined over O , i.e., a Hilbert modular form HaliftK p defined over OK such that Haliftp (A,↵) ⌘ Hap(A⇥OK O1,↵⇥OK O1) (mod m). Unfortunately, this is not always possible. However, we are always able to produce such a lift of a power of the Hasse invariant Hawp for some positive integer w. We will refer to this lift as Haw,liftp . Note that an HBAV A defined over OK is ordinary at p if and only if Haw,liftp (A,↵) is a unit for some choice of ↵. In addition, because of the transformation property that comes with Haw,liftp being a Hilbert modular P form of weight 2 w(q 1), the p-adic valuation of Ha w,lift p (A,↵) does not Ip depend on the choice of ↵. Define the Hodge height Hdgp for each prime p, ⇢ 1 Hdgp(A) = min (Ha w,lift vp p (A,↵)), 1 . w The condition that A is ordinary at p if and only if Haw,liftp (A,↵) is a unit translates to the condition that A is ordinary at p if and only if Hdgp(A) = 0. Consider a tuple v = (vp)p of rational numbers vp 2 (0, 1)\Q, indexed by the primes p. For each such tuple, we construct the v-overconvergent locus X(v), which is a normal formal scheme that classifies HBAVs A with Hdgp(A) < vp for all p. This description gives a well-defined subset of the rigid analytic fiber X (v) ⇢ X ; we take a formal model X0(v) for it by taking admissible formal blowups as in [AIP16, Section 3.2], and we normalize the resulting formal scheme in X (v) to obtain the formal overconvergent locus X(v). We also consider the formal completion of the ordinary locus Xord inside X; since an ordinary HBAV has Hdgp(A) = 0 for all p, we have an inclusion Xord ! X(v) for any tuple v. For each tuple v with 1vp < n for all p, the canonical subgroup of level npp on Xord extends to a canonical subgroup Cn ! X(v) over the v-overconvergent 53 locus. Its fiber over the ordinary locus is locally isomorphic to the formal group scheme d1 ⌦ ordµpn . Its fiber over the complement X(v) \ X is isomorphic to a di↵erent formal group scheme, but both become isomorphic to each other and to the constant group scheme O nF/p OF at the rigid fiber. We move to the rigid fiber Cn ! X (v) in order to define In = Isom 1X (v),F (d ⌦ µpn , Cn), the group of OF -linear isomorphisms between d1 ⌦ µpn and Cn. The forgetful map hn : In ! X (v) is a finite étale cover with Galois group AutF (d1 ⌦ nµpn) = T(Z/p Z). We then let In be the formal model of In obtained by normalizing X(v) inside In, and let hn also refer to the covering In ! X(v). We call In and In “partial Igusa towers”. Remark 4.4.1. Here we are really using the fact that the fibers of the rigid space Cpn are all isomorphic, even though the fibers of the formal model Cpn are not. This partial Igusa tower In is supposed to be the overconvergent version of the full Igusa tower T⇥C 1 from Section 4.3, classifying trivializations of the canonicalp subgroup. Over the rigid fiber X (v), In literally does this. However, this becomes more complicated over the special point when the fibers of Cpn are no longer all isomorphic to each other. The canonical subgroups help us pick out an important subsheaf F ⇢ !, which is described in [AIP16, Proposition 3.4]. Proposition 4.4.2. There is a unique subsheaf F of ! which is locally free of rank supp{vp} 1 as a OF ⌦Z OIn-module and contains p p1 !. Moreover, for all 0 < w < n nsup { } pp vp 1 , we have OF -linear mapsp HTw : C D ⇤ w ⇤ pn ! h (F)/p h (F).n n 54 These induce isomorphisms D w ⇤ w ⇤ HTw ⌦ 1: C n ⌦ (Op In/p OIn)! h (F)/p h (F). (4.7)n n Sketch. We sketch the proof, referring to [AIP16, Proposition 3.4], and [AIP15, Proposition 4.3.1], for a full proof. The maps HTw and the sheaf F are constructed at the same time, and then we check that HTw has the required properties of being OF -linear and inducing the isomorphism in Equation (4.7). Let A(v) = A ⇥X X(v) be the universal object over X(v). Pick any basis 0 0e1, . . . , e of the rigid group d 1 ⌦ µpn , and for each pointd (A, u) of In we get a basis 1e1, . . . , ed for ⌦C n by ej = Du ( 0e ) 2 CD ⇠ 1j pn = ⌦p CD . Recallpn that the inclusion Cpn ! A(v) induces a pullback ⌦1 1( ) ! ⌦C n . Lift each ej to anA v p element ẽj 2 ⌦1 ( ) such that ẽj maps to ej for each j. The sheaf F is the subsheafA v of ⌦1 ( ) generated by ẽ1, . . . , ẽd. The map HTw is determined by sending eA v j to ẽj for each j. In loc. cit. the authors give the details of the construction. supp{vp} Remark 4.4.3. The fact that ! ⇢ F ⇢ p p1 ! shows that F ⌦OK K = ! ⌦OK K as families of modules. Their rigid fibers give di↵erent integral structures to the rigid analytification of !, and only F gives a locally free sheaf of OF -modules. Specifically, the rigid fiber of ! and the rigid fiber of F should be thought of as two versions of “the ball of radius 1” inside the analytification of ! pulled back to In. 4.5 Overconvergent Hilbert Modular Forms Fix a tuple v such that 1vp < n for all i. We have our canonical subgroupp Cpn ! X(v) of level np and a partial Igusa tower In. Write evD1 : In ! CD for the map sending Du to u (1). This descends to an map of formal schemes evD1 : I ! CDn by the universal property of the relativen normalization. Following [AIP16], define the formal ane morphism ⇥w : TF !,w 55 n In, for any n 1 < w  n sup{vp} p1 , as follows. For every normal p-adicallyp complete and separated, flat OK-algebra R and for every morphism : Spf(R) ! In, its R-valued points over classify frames ↵ : ⇤ (F) ! OF ⌦Z R, such that HTw(evD 11 (u)) ⌘ ↵ (1) (mod wp ). The reference cited above writes this as needing to send 1 to 1 in the composite D O nO uF/p F ! CD(R) HT!w ⇤ F w ⇤/p (F) !↵ O wF ⌦Z R/p R. (4.8)n The map w : T ⇥ F ! In is a formal torsor for the group T0 . Composing with the,w w projection In ! X(v) gives a formal torsor pr : T⇥F ! X(v) for the group T .w ,w w The rigid fiber is denoted T ⇥F .,w Remark 4.5.1. In [AIP16], this is called IW+. In other references, such as [AIP15, w Liu19a], there are two related formal torsors: one is called IW+, while the other is w T⇥F . In the case of Hilbert modular forms, the two coincide – we use the second,w name here so that our discussion in Section 4.6 better parallels that of [Liu19a, Section 3.4]. For any representation (,W) of Tw, we may construct a corresponding sheaf ! on X (v). Over an anoid open U = SpR ⇢ X (v) when !|U is trivial,,w sections of ! correspond to functions ,w 0 H (X ( ) ⇥ 1v ,! ) = f : TF (U)! W ⌦R | f(A,↵g) = g · f(A,↵) .,w ,w Definition 4.5.2. Let  be a w-analytic weight. A v-overconvergent Hilbert modular form of level 1(N) and weight  is a section of ! defined over X (v).,w The K-vector space of v-overconvergent Hilbert modular forms of level 1(N) and weight  is thus 0H (X (v),! ). ,w 56 An overconvergent Hilbert modular form is a v-overconvergent Hilbert modular form for some tuple v with each vp > 0. The K-vector space of S overconvergent Hilbert modular forms is the union 0H (X (v),! ). v ,w Remark 4.5.3. In Remark 4.3.2, we saw how to view classical Hilbert modular forms as p-adic Hilbert modular forms. The idea is that, over the ordinary locus, there is a tight connection between the canonical subgroup of the universal ordinary HBAV Cp1(Aord) and the Hodge bundle !. Since overconvergent Hilbert modular forms are p-adic modular forms, we would like to use the canonical subgroup when describing them. However, the full canonical subgroup Cp1 does not extend past the ordinary locus at all – if we want to use the canonical subgroup, we have to extend one of the finite level canonical subgroups Cpn . The sheaf F fills in the extra information that we would otherwise lose by not having the full canonical subgroup of infinite level. The compatibility condition, requiring that 1 be sent to 1 in Equation (4.8), essentially means that the trivialization u of Cpn controls as much as possible about the trivialization ↵ of F . 4.6 Nearly Overconvergent Hilbert Modular Forms and the Main Construction There is a natural homomorphism Q ! T that picks out the top left entry. Let 0Q be the rigid analytic group which is the preimage of T0 under this w w projection, and Qw the preimage of Tw. We have previously considered the OK-schemes ⇥T = IsomX(OF ⌦! Z OX ,!) and ⇥,+ + 2TH = Isom ([OF ⌦Z OX ] ,H), where the + superscript means that theX isomorphisms should respect the filtrations. We write T ⇥ and T ⇥,+H for their rigid!,an ,an analytifications, and T ⇥ ( ) and T ⇥,+v H (v) for their base changes to X (v). We have!,an ,an 57 a map T ⇥,+ ⇥H (v) ! T (v) given by forgetting everything but the isomorphism on,an !,an the submodules OF ⌦Z OX (v) and !. Then let T ⇥F ! Iw ! X (v) be the rigid fiber of the chain of morphisms,w defined in Section 4.4. This T ⇥ ⇥F maps to T as well.2 We thus form the fiber,w !,an product T ⇥,+H (v) := T ⇥,+( ⇥v)⇥ ⇥ T (v). ,w H,an T!,an(v) F ,w The map +⇡ : T ⇥,+H (v)! X (v) is a torsor for the group Qw ,w w. Let  be a w-analytic weight. There is an associated rigid analytic representation W,w of Tw, which we inflate to Qw. In similar fashion to the holomorphic case, we define the analytic (g, Qw)-module V,w by V,w = W,w ⌦OH\Q We give g(K) the same basis as before; since Qw is open in Q, its Lie algebra is still q ⇢ g. We define the actions of g(K) and Qw(K) by the same formulas, 0 1 Ba b· C(g P )(Y ) = a · P ( 1a (b+ Y d)) for all g = @ A 2 Qw(R), 0 d ( µ · 2 @P )(Y ) = Y" · P (Y ) Y P (Y ). @Y Compatibility is checked in the same way. Since the action of Qw preserves the degree of each Y, there is a Qw-submodule ⌫V ⇢ V,w consisting of,w polynomials with degree at most r in the variable Y for each . Since g 2It does not map to the rigid fiber of the formal completion of ⇥T! , as the points of either space should be OF ⌦Z OK-bases for the respective sheaves, which do not match up. The points of T ⇥!,an are instead OF ⌦K-bases for !. 58 preserves the degree as a function of Y⌧ for each ⌧ =6 , and raises the degree as a function of ⌫ ⌫+Y by at most 1, we have gV ⇢ V .,w ,w Definition 4.6.1. The sheaf of nearly v-overconvergent Hilbert modular forms of level 1(N), w-analytic weight , and type is V⌫ ( ) = T ⇥,+ ⌫ ⌫ v H (v) ⇥ V . A,w ,w Qw ,w nearly overconvergent Hilbert modular form of level 1(N), w-analytic weight , and type ⌫ is a section of V⌫ (v) for some v with each vp satisfying 0 1< v,w p < pw . Proposition 3.4.4 gives us a connection r 1,w : V,w ! V,w ⌦ ⌦X ( )(logC),v and the Kodaira–Spencer morphism gives us our di↵erential operators, r,w : V,w ! V+2,w The discussion of Remark 3.5.4 applies, giving r ⌫ ⌫+,w : V ! V +2 .,w  ,w Each r,w commutes with r⌧,w for any pair of embeddings , ⌧ 2 I. Thus we may unambiguously write r0,w for the di↵erential operator that raises weights 0 Pby = 2 0 k , and types by 0⌫ = ( 0k ). We can also base change to X ( 0v ) for any 0v = ( 0vp) with 0 0 < vp < vp for each p | p. This construction works just as well for a single weight  as it does for the universal w-analytic weight un , and pulling back by the inclusion of an w anoid open U ⇢ Ww allows us to apply these di↵erential operators to families parametrized by U . We summarize these results in the following theorem. Theorem 4.6.2. Fix a tuple v = (vp) with 0 1 < vp < w for all i. For eachp embedding : F ! K, and any k 1, there is a di↵erential operator rk acting on families of nearly v-overconvergent Hilbert modular forms of w-analytic 59 weight, which raises the weight by 2k and the type by k. The operators rk and r` ⌧ commute for any pair of embeddings and ⌧ . 4.7 Integrality As constructed, this di↵erential operator is defined over K. However, when p is unramified in F , it is defined over OK . Each module ⌫V is defined over Z, but our description and the basis we chose are only defined over K a priori. It is true that the connection r is defined over OK in general; however, there is a problem to defining r when p is ramified in F . L Over K, we can decompose g = g. Any trivialization ↵ 2 ⇥,+TH over an ane open U = SpecR defined over K determines a trivialization of !, and thus of M ⌦2 ⇠= ⌦1! (logC). X We let D 2 TX(U) be the direction dual to that trivialization, and X(D,↵) 2 g(R) such that r(D)(v)(↵) = (X(D,↵) · v) (↵). In fact, we get a basis _D for each ⌦2! ; we embed that sheaf as a summand of ⌦1 (logC) and view the set { _D } as a basis X of the latter sheaf. Let D 2 TX(U) be dual to _D . We have X(D,↵) 2 g(R), P P and D = D giving X(D,↵) = X(D,↵). Specifically, X(D,↵) 2 g(R) L corresponds to the tuple (X(D,↵)) 2 g(R). Under this formalism, we have r(v)(↵) = d(v)(D)(↵) + (X(D,↵) · v) (↵). With this in mind, we turn to integrality. When p is unramified in F , the L decomposition g = g manifests over OK , and r acts integrally. However, when p ramifies, there is no guarantee that X(D,↵) 2 g(OK) under the assumption that X(D,↵) 2 g(OK). However, we can quantify by how much this fails, and we do so using the following lemma, formulated and proved by the author for the discussion in this section. 60 Q Lemma 4.7.1. Let n = (n) 2 OK. There exists some ` not depending on n Q such that `p n 2 OF ⌦OK ⇢ OK. Proof. Pick a Z-basis {e1, . . . , ed} for OF , and order the embeddings of F into K, Q {1, . . . , d}. The OK-linear map OF ⌦ OK ! OK is given in this basis by the matrix 0 1 B1(e1) . . . d(e1)B C M = BB .. .. . . . .. C C @ CA 1(ed) . . . d(ed) It is classical that det 2M = is the discriminant of F/Q. Further, the index of the image of this OK-linear map is #OK/(detM). Let p be the maximal ideal in OK and write (detM) = p` for some `. We have #O p`K/ = f`p , where f is the residue Q field degree. Thus f`p (n) 2 OF ⌦ OK whenever (n) 2 OK . This exponent does not depend on the choice of n, so the lemma is proven. Q Remark 4.7.2. In fact, we can say more. Since ( OK) /(OF ⌦ OK) is a OK- module of order f`p , it is killed by `p - the worst case scenario is that the quotient is indecomposable, and thus isomorphic to the indecomposable module O `K/p . This module has size f`p , but is killed by `p . We use this to prove more about the Lie algebra g. Q Corollary 4.7.3. Let X = (X) 2 g(OK) ⇢ gl2(OK) ⇢ g(K). View X Q as the tuple (0, . . . , X, . . . , 0) in gl2(OK) ⇢ g(K). Then there exists some ` independent of X such that `p X 2 g(OK). Proof. Pick ` as in Lemma 4.7.1. Note that the elements of g(OK) should be viewed as 2 ⇥ 2 matrices with entries in OF ⌦ OK , while X is a priori a tuple of 2 ⇥ 2 matrices with entries in OK . We may instead view X as a 2 ⇥ 2 matrix 61 Q with entries in OK . Scaling `X by p simply scales its entries, so by our choice of the entries of `` p X are in O `F ⌦OK , and p X 2 g(OK). By the corollary, we have `p X(D,↵) 2 g(O `K). Thus p X(D,↵) acts integrally. ( ) · 1( ⌫(O )) ⇢ ⌫+(O ) ⇢ ⌫+X D,↵ V K V K V (K).`  p On the level of sheaves, the above translates to r 1(V⌫(O )) ⇢ V⌫+ K (O ⌫+ +2 K) ⇢ V (K).` +2p This allows us to control denominators. 62 CHAPTER V DESCENT TO G 5.1 Two Groups The phrase “Hilbert modular form” is ambiguous in that it can refer to the space of automorphic forms on one of two groups. The group G = ResO /Z GL2 isF one. For any commutative ring R, its R-points are G(R) = GL2(OF ⌦Z R). Determinants of such matrices live in T = ResO /Z Gm. Recall that over K, T is aF split torus; its diagonal subgroup is isomorphic to Gm. The second group, denoted ⇤ G , is defined as the fiber product ⇤G = G ⇥T Gm. It is a subgroup of G, with R-points ⇤ G (R) = {g 2 G(R) | det(g) 2 1⌦ ⇥R ⇢ (OF ⌦R)⇥}. Each group has advantages. The Hecke theory for G is canonical, while the theory for ⇤G is not. In fact, unlike ⇤G, G has a noncommutative Hecke algebra. On the other hand, while both are the groups of interest in a PEL type moduli problem, only the moduli problem associated to ⇤G is representable by a scheme. This is the reason why the previous sections were only concerned with the automorphic forms on ⇤G . Since ⇤G is a subgroup of G, we may restrict the automorphic forms on G to the group ⇤G , viewing them as automorphic forms on the latter group. This allows us to use the geometric viewpoint of the previous sections to study automorphic forms on G. In the next section, we describe the weights of such form, and following [AIP16] we record the geometric condition that distinguishes these restricted forms from other automorphic forms on ⇤G . 63 5.2 Weights In Section 2.1, we introduced the weight space for overconvergent modular forms on ⇤G ; here we continue with the weight space for G, including the weight of a restricted modular form. The space of algebraic weights for G is {characters of T/K} ⇥ Z, with a map to the algebraic weights for ⇤G given by ⇢ : (✓, w) !7 2✓ + wNmF/Q. In particular, if (✓, w) is a weight for G, ⇢(✓, w)(g) = ✓(g)2 · (NmF/Q(g))w for any g 2 T(Zp). If we have a modular form of weight (✓, w) on G, its restriction to ⇤G will have weight ⇢(✓, w). The space of p-adic weights for G is {characters of T(Zp)} ⇥ Zp. The map ⇢ sending a p-adic weight for G to a p-adic weight for ⇤G uses the same formula as above. Let f be an automorphic form on ⇤G of weight 2✓ · Nmw Q. It can beF/ extended to an automorphic form on G if, for all 2 O⇥,+✏ , we have F f(A, ◆, , ✏,!) = ✓(✏)f(A, ◆, ,,!). (5.1) We will avoid using this criterion. It is included to stress the fact that whether or not an automorphic form on ⇤G extends to G can be detected geometrically. We will opt instead to use a criterion involving the symmetric space. This will be enough, as Proposition 5.4.1 can be stated in terms of classical Hilbert modular forms and easily transported to the space of overconvergent ones. 5.3 Descent of Modular Forms We can quantify the di↵erence between automorphic forms on ⇤G and automorphic forms on G using the adèlic viewpoint. Let +K1 be a maximal compact subgroup of the connected component of the identity G(R)+ in G(R), and ⇤ K1 the same for ⇤ G (R). Let K0(N) and ⇤K0(N) be the natural choices of compact 64 open subsets of the finite adèles, consisting of matrices which are upper triangular modulo N . The inclusion ⇤G ! G induces a map on the symmetric spaces ⇤ G (R)+ ⇤ ⇤ + +/K1Z(G (R)) ! G(R) /K1Z(G(R)) which ends up being a bijection. Thus automorphic forms on G and automorphic forms on ⇤G are functions on the same space. The only di↵erence comes from the congruence subgroups 80 1 9 <> >a b = ⇤( ) = ⇤(Q) \ ⇤ B0 N G K0(N) = >@ CA 2 SL2(OF ) | c ⌘ 0 (mod N) , : >c d ;80 1 9 > > (5.2)@ A 2 GL + 2 (OF ) | c ⌘ 0 (mod N) : > . c d ; 0 1 B✏ 0C For any ✏ 2 O⇥,+, let g✏ = @ A. Equation (5.1) is simply a translationF 0 1 of the fact that Gg✏ 2 0 (N), while g✏ 62 ⇤0(N). In a strict sense, this is the only discrepancy, as ⇤0(N) and these matrices g✏ together generate G 0 (N). We define an action of O⇥,+ on the space of automorphic forms on ⇤G of F weight ⇢(⌫, w) to be ✏ · f = f |g✏. (5.3) The discussion above implies that f is a modular form for G if and only if O⇥ acts F on f via its nebentypus. 5.4 Descent of Operators At the moment, the di↵erential operators r are defined as maps that send Hilbert modular forms on ⇤G to Hilbert modular forms on ⇤G . A priori, if f is a nearly overconvergent Hilbert modular form for G, rf is only a nearly overconvergent Hilbert modular form for ⇤G . We will argue that it extends to G. 65 Proposition 5.4.1. The action of O⇥,+ defined in Equation (5.3) commutes with F the Gauss–Manin connection r : H! H⌦ ⌦1 . X Proof. This is classical, and seen e.g. as a special case of [Shi00, Proposition 12.10(2)]. For another proof, we note that the slash operator is defined in such a way that it commutes with the exterior derivative d defined for di↵erential forms. Since r arises as a di↵erential in the spectral sequence associated to a filtered de Rham complex whose di↵erentials are this d, it commutes with the slash operator as well. This leads to our second Main Theorem. Theorem 5.4.2. The di↵erential operators r constructed in Theorem 4.6.2 preserve the space of Hilbert modular forms for G inside the space of Hilbert modular forms for ⇤G . Proof. This is essentially a special case of Proposition 3.4.4. The fiber of ⇥,+TH over some point x corresponding to a HBAV (A, ◆, ,) only depends on the substring (A, ◆), so we may identify the fibers over any x with the fibers over g✏ · x. Under this identification, the slash operator is just the natural action of 3 on V⌫Q g✏ , L and we proved that this action commutes with r = r in the previously cited proposition. Thus the fact descends to each r. 66 APPENDIX DAMERELL’S FORMULA This text is adapted from a note published on the author’s website. It is independent from the rest of the dissertation, but is included to show the arithmetic application that motivated the construction of the di↵erential operators in the main body of the text. A.1 Introduction In [Ser72], Serre gave a construction of the Kubota–Leopoldt p-adic zeta function using the theory of p-adic of modular forms and the fact that the values of holomorphic Eisentein series at the cusps are related to the values of the zeta function. This construction was generalized in [DR80] to construct the p-adic zeta functions of totally real fields. These two “interpolation” results were preceded by “algebraicity” results: up to a renormalization, the values of these zeta functions at certain inputs are in fact algebraic numbers, so that it makes sense to ask about congruences between them. For quadratic imaginary fields, this study was initiated by Damerell, and completed using the Maass–Shimura operators and the theory of nearly holomorphic modular forms. From here, p-adic interpolation of the zeta functions of quadratic imaginary fields, and then for CM fields,1 were given by Katz in [Kat76] and [Kat78] respectively, though with conditions on p. Here, we make explicit the relationship between values of Eisenstein series at CM points and values of zeta functions for CM fields. We begin by describing the L-functions to be interpolated in Section 2, introducing Hilbert modular forms 1Recall that a CM field is a totally imaginary field K which is a degree 2 extension of a totally real field F . 67 in Section 3. Once we have both tools, we relate them in Section 4. Section 5 gives some reasons one might care about the result. An excellent technical reference on the theory of nearly holomorphic automorphic forms is [Shi00], which summarizes and builds on earlier results of Shimura and Maass. Katz’s construction of p-adic L-functions has been extended in the case of quadratic imaginary fields by Andreatta and Iovita in [AI19]; the author of the present note is working to extend this to the case of CM fields. A.1.1 Notation, Conventions, and Assumed Knowledge. We assume the reader has taken a first course in algebraic number theory (e.g. out of [Mil20]) and has a basic understanding of complex analysis. A.2 L-functions For this section, we fix a totally real field F of degree d, and a CM extension K = F (↵). For simplicity, we assume that the ring of integers is O 2K = OF + ↵OF . We also fix a CM type of K; for each real embedding : F ! R, we choose a preferred embedding of K into C which agress with when restricted to F , which we also call . Thus the set of complex embeddings of K is the set of ’s and all ’s as runs over the real embeddings of F . Write I for the set of real embeddings of F . A.2.1 The 1-variable L-function. The standard, 1-variable Dedekind zeta function of K can be written, for Re(s) > 1, as the sum over all nonzero ideals of OK . X 1 ⇣K(s) = . (A.1) Nm(a)s a 2If F is a PID, this is possible because K/F is quadratic. If not, this decomposition should be OK = OF + ↵a for a fractional ideal a of F . 68 We may rewrite this in order to make most of the summation happen with elements of the field itself. Pick representatives a1, . . . , ah for the class group of OK , and note that every integral ideal a can be written as a = a1(↵) for some i and some ↵ 2 ai.i Writing a as a1(↵) for some i and some ↵ 2 ai is unique up to rescaling ↵i by a unit in O⇥ . In the quadratic imaginary case, we can sum over all elements K of ai, and divide by #O⇥ to account for repetition. When K is not quadraticK imaginary, its unit group is infinite, and we can no longer do this. We should instead view a as a O⇥i -set, where u · ↵ = u↵ for any unit u and ↵ 2 ai. ThisK action is useful because each orbit ⇥↵O 2 ai/O⇥ corresponds to a unique integralK K ideal a1(↵), and each integral ideal a1(↵) corresponds to a unique orbit ↵O⇥ . i i K In fact, for reasons that will arise later, we will look at orbits for the restricted action of O⇥,+ ⇢ O⇥ consisting of totally positive units u with (u) > 0 F K for every real embedding of F . The association O⇥,+ !7 a1↵ (↵) is then F i [O⇥ : O⇥,+]-to-one, where [O⇥ : O⇥,+] is the index of the subgroup O⇥,+ K F K F F in O⇥ .3 Thus we write the following, where the innermost sum is over cosets K 0 6= O⇥,+↵ 2 a ⇥,+i/O .F F 1 Xh X Nm(a )s ⇣K(s) = i ⇥ ⇥ + . (A.2)[O : O , ] Nm(↵)s K F i=1 O⇥,+↵ F A.2.2 Hecke Characters. We will give a definition in terms of Hecke characters in order to write down the (d + 1)-variable zeta function in all cases. However, in order to avoid delving into the idèlic theory, we give the following ideal-theoretic definition of a Hecke character, instead of the standard one. 3This index is finite. In particular, [O⇥ : O⇥,+F F ]  2d, so that O ⇥,+ F is a finitely generated Abelian group of the same rank as O⇥. Since O⇥ also has the same rank as O⇥, so O⇥,+ ⇢ O⇥F K F F K is an inclusion of finitely generated Abelian groups of the same rank. The quotient is finitely generated of rank 0, hence finite; its size is the index we’re looking for. 69 Definition A.2.1. An unramified Hecke character is a homomorphism : IK ! C⇥ from the group of fractional ideals of K to the complex numbers. An unramified Hecke character is unitary if |(a)| = 1 for all ideals a 2 IK . We say that has infinity type (k1,, k2,) if can be written as a product of characters Q = k 1, where is unitary and 1((↵)) = (↵) 1, k(↵) 2,u u for all principal ideals (↵). Remark A.2.2. Unramified, unitary Hecke characters have the property that ((↵)) = 1 for all principal ideals (↵). We note that ramified Hecke characters exist. For simplicity, we only work with unramified Hecke characters, but some of what we say will be true for all Hecke characters. We distinguish these statements and constructions by writing “(unramified) Hecke character” when the unramified hypothesis is not needed. In particular, we note that the map a 7! Nm(a)s is an unramified Hecke character of infinity type (s, s). For any (unramified) Hecke character , write: X (a) X 1 L( ) = ( 1, s , L , 0) = . Nm(a)s (a) a a When s 1 = Nm , we see that L( , 0) = ⇣K(s). More generally, L(Nm k , s) = L(, s k) for any (unramified) Hecke character and any two complex numbers k and s. A.2.3 The (d+1)-variable L-function. We build the (d+1)-variable L-function following Equation (A.2). Again, the innermost sum is over cosets 0 6= ↵O⇥ 2 ai/O⇥.F F Xh1 X (a ) L(, (s, t)) = i ⇥ ⇥ Q . (A.3)[O : O ] (↵) (↵)s(↵)t K F i=1 ⇥ ↵OF 70 Note that L(, s) = L(, (s, s)), and changing the CM type at (i.e., replacing the preferred choice of complex embedding by ) interchanges the variables s and t. Remark A.2.3. For this to be well-defined, we need the sum s + t to be independent of , since ↵ is only defined up to a totally positive unit in F . Write S0 for this common value. If we replace ↵ by ⇥,+ u↵ for some u 2 O , (u↵) = F (u)(↵) does not change; since is unramified and unitary, we have (u) = 1. However, we must also check that Y Y s t s t (u↵) (u↵) = (↵) (↵) . (A.4) Since (u) = (u), the left hand side di↵ers from the right hand side by a factor of ! Y Y Y S0 ( su) +t = S(u) 0 = (u) = Nm( )Su 0 = 1. Thus we have verified the equality in Equation (A.4), and so we see that the function defined in Equation (A.3) is independent of the choices of the representatives 2 O⇥,+↵ ↵ . F This is why we describe it as a (d + 1)-variable L-function when it looks like there are 2d variables - once the common sum S0 = s + t is chosen, the choice of s forces the choice of t, and vice-versa. Thus we are left with d + 1 variables; one way to choose these variables is to vary S0 and the s’s, though there is no preferred way to choose. A.3 Hilbert Modular Forms For now, we focus on the totally real field F of degree [F : Q] = d. There are many ways of viewing Hilbert modular forms over F . In this section we describe three. (a) Hilbert modular forms as functions on the space of lattices, 71 (b) Hilbert modular forms as holomorphic functions on a symmetric space, and (c) Hilbert modular forms as sections of a line bundle on a moduli space of Abelian varieties. The first description will be important for relating values of certain Hilbert modular forms to the values of the (d+ 1)-variable L-function. The second is where the Maass–Shimura operators will be described. The third will be important for applications to algebraicity. A.3.1 Setup. Fix a set of representatives a1, . . . , ah for the class group of F . Note that this also serves as a complete set of isomorphism classes of locally free OF -modules of rank 1. Each fractional ideal ai lives naturally in F . Since R is a flat Abelian group, Q we may also view ai as a subset of F ⌦Z R ⇠= R, where the isomorphism sends Q a pure tensor n ⌦ t 2 F ⌦ R to the tuple ((n)t) 2 R. Each ai ⇢ F ⌦ R is a discrete subgroup; we assign a volume to the quotient (F ⌦ R)/ai by choosing a fundamental domain 4D , and defining the volume of the quotient to be the volume of this subset of F⌦Z R. We also refer to this as the covolume of the lattice ai. We scale the representatives ai for the class group. Let Vi denote the covolume of ai in F ⌦ R, and the discriminant of F . Replace ai by the latticep 2d || pd ai ⇢ F ⌦ R; this need not be a fractional ideal of F , but it is a lattice Vi p in F ⌦ R with covolume ||. In particular, the choice of ai is unique, and the representative of the class consisting of principal ideals is OF . A.3.2 Lattices. We describe Hilbert modular forms as functions on a space of lattices. First, we should define what a lattice is. 4 D is a connected subset of F ⌦ R such that, for all x 2 F ⌦ R, exactly one element d 2 x + ai is in D. 72 Definition A.3.1. Consider the vector space F ⌦Z C. It has the structure of a OF - module by acting on the first component, and a real vector space by acting on the second component. A lattice, or more precisely an OF -lattice in F ⌦ C, is a discrete OF -submodule L of F ⌦ C which is locally free of rank 2 as an OF -module, and which spans F ⌦ C over R. Since a lattice L is locally free of rank 2 as an OF -module, it is isomorphic to ai1 ai2 for two indices i1 and i2, where ai refers to one of the representatives of the class group of F chosen above. Picking an isomorphism, we may write L = !1ai1 !2ai2 for some !1,!2 2 (F ⌦ C)⇥. We say two lattices 0L and L are homothetic if there is some 2 (F ⌦ C)⇥ for which 0L = {` 2 F ⌦C | ` 2 L} = L . By scaling L = !1ai1 +!2ai2 , we see that any lattice is homothetic to a lattice of the form a ⇥i1 + ⌧ai2 for some ⌧ 2 (F ⌦ C) ; e.g., for !⌧ = 2 . !1 We now define Hilbert modular forms in terms of functions on lattices. Definition A.3.2. Let L be the set of OF -lattices in F ⌦ C, and let k = (k) be a tuple of integers indexed by the real embeddings of F . A Hilbert modular form of weight k is a function f : L! C satisfying some analytic conditions (to be specified in the next section) and the homogeneity property ! Y f(L) = ()k f(L) for all 2 (F ⌦ C)⇥. Here, if = n⌦ z 2 F ⌦C is a pure tensor, we write () = (n)z. It is nonzero for all ⇥ if 2 (F ⌦ C) . Notice that f is determined by its values on lattices of the form ai1 ⌧ai2 . 73 Remark A.3.3. In Section A.4.1, we will give our first example of a Hilbert modular form using this viewpoint. This will be the easiest way for us to relate its values to the values of the L-functions. A.3.3 Holomorphic Functions. Hilbert modular forms can also be viewed as holomorphic functions on a symmetric space. Let h ⇢ C be the upper half-plane consisting of complex numbers with positive imaginary part. Then define hF to be the product of d copies of h, indexed by the real embeddings of F . We write z = (z) for elements of hF . We have an action of the group SL F on h given by the formula 0 1 2 F ✓ ◆ Ba bC · (a)z + (b)@ A (z) = . (c)z + (d) c d This action gives us the following definition. Definition A.3.4. A holomorphic Hilbert modular form of weight k = (k) is a holomorphic function f : hF ! C such that |f(z)| is bounded as every Im(z) goes to 1 at once5 and which satisfies the transformation property ! 0 1 Y · B a bC f( z) = ((c) kz + (d)) f(z) for all = @ A 2 SL2 OF . c d To connect this with the previous definition, we have to associate a lattice to each z 2 hF . In fact, the space of lattices is disconnected, while hF is connected – a modular form in the sense of Definition A.3.2 is actually a collection of Hilbert modular forms in the sense of Definition A.3.4, one for each connected component. Thus we should actually associate multiple lattices to each element of hF . View z 2 hF as an element of F ⌦ C hF . To every triple (ai1 , ai2 , z), we associate a lattice ai1 + zai2 = {a1 + a2z 2 F ⌦ C | aj 2 aij}. This allows us to 5This condition is important for F = Q and automatic for d > 1. 74 translate between Definitions A.3.2 and A.3.4. Note that the same lattice may be written in di↵erent ways. For a Hilbert modular form fL viewed as a function on the space of lattices, we build a function fh(ai1 , ai2 ,) : hF ! C for each orered pair (ai1 , ai2) by the formula fh(ai1 , ai2 , z) = fL(ai1 +zai2). One may check that this function satisfies the requisite transformation property. The analytic conditions mentioned in Definition A.3.2 correspond to the function fh(ai1 , ai2 ,) being holomorphic and bounded at the cusps. Now, for each ordered pair (ai1 , ai2), fix a holomorphic modular form fh(ai1 , ai2 ,) : hF ! C. Every lattice !1ai1 + !2ai2 is homothetic to ai1 + zai2 for some z 2 hF . Specifying that fL(ai1 + zai2) = fh(ai1 , ai2 , z), the homogeneity property then determines fL on all lattices. Remark A.3.5. Viewing Hilbert modular forms as functions on the space of OF - lattices in F ⌦ C will give a tight connection to the relevant values of L-functions. However, one of the important pieces of the theory, the Maass–Shimura operator, is best understood in the realm of holomorphic functions. We give both viewpoints here, and a rough translation between them, in order to take advantage of both sides of the theory. The third viewpoint, in terms of algebraic geometry, is given for the purpose of one of our applications. The next section introduces this, and ties it back to the viewpoint involving lattices. A.3.4 Geometry. In order to motivate the following discussion, we give a preliminary definition of classical modular forms; they may be viewed as a special case of Hilbert modular forms over the totally real field F = Q. Definition A.3.6. Fix a base ring R. A modular form defined over R is an algebraic function f that assigns to every pair (E,!) of an elliptic curve E defined 75 over some R-algebra S and a basis ! for ⌦1 as an S-module, an element of S. E/S A modular form of weight k is such a function which satisfies the homogeneity property that for any ⇥c 2 S , ( kf E, c!) = c f(E,!). Remark A.3.7. Since ⌦1 is a free S-module of rank 1, any two bases ! E/S 1,!2 for ⌦1 are related by !1 = c!2 for a unique ⇥c 2 S . Thus the value of f(E,!) forE/S any ! is determined by the value of f(E,!0) for some fixed !0. Elliptic curves are one-dimensional examples of Abelian varieties. In order to generalize the definition above, one may look at more general classes of Abelian varieties of higher dimension, or endowed with extra structure. For Hilbert modular forms, we consider Hilbert–Blumenthal Abelian varieties. Definition A.3.8. Fix a ring R, and let F be a totally real field of dimension d. A Hilbert–Blumnenthal Abelian variety, or HBAV, is a tuple (A, ◆, ,) consisting of – an Abelian variety A of relative dimension d over R, – a real multiplication ◆ : OF ! End(A), – a level structure , and – a polarization . For our purposes, the level structure is trivial, and we suppress it from the notation. Remark A.3.9. In general, the moduli space of HBAVs is represented by a stack, rather than a scheme. However, with enough level structure, the moduli space is “rigidified” to be represented by a scheme. When we do not fix a polarization as part of the data, this never happens. For our purposes, we will have no level 76 structure, which will mean that we have to deal with stacks; we will ignore the issues that come up. The real multiplication ◆ gives ⌦1 the structure of an OF ⌦ R-module. ForA/R a dense open subspace of the moduli space of HBAVs, this is a free module of rank 1. In this case, we can form its frame bundle consisting of bases for ⌦1! which A/R generate it as an OF⌦R-module. Any two such bases !1,!2 are related by !1 = c!2 for a unique c 2 (OF ⌦R)⇥. Definition A.3.10. Fix a base ring R. A Hilbert modular form defined over R is an algebraic function f that assigns to every tuple (A, ◆,,!) of an HBAV (A, ◆,) defined over some R-algebra S and a basis for ⌦1! as an OF ⌦ S-module, anE/S element of S. A Hilbert modular form of weight (k) is such a function which satisfies the homogeneity property that for any c 2 (O ⇥F ⌦ S) , f(A, ◆,, c!) = Q ( )kc f(A, ◆,,!). Here acts on pure tensors ⇥n ⌦ s 2 (OF ⌦ S) by (n⌦ s) = (n)⌦ s. The algebraic geometry involved gives a way to investigate algebraicity, and in fact the integrality, of the values of Hilbert modular forms. The modular forms we consider will be defined over Z. Thus the algebraicity of their values will depend on the algebraicity of the inputs at which we evaluate them; the integrality will depend on those inputs and on the structure of the moduli space of HBAVs viewed as a stack, which will contribute some predictable denoninators. We connect this section back to the theory of lattices. Let (A, ◆,) be an HBAV defined C, and let ! 2 ⌦1 be a basis. The choice of ! gives an A/R isomorphism between Lie algebra Lie(A) = H1(A,R) and F ⌦ C, by sending a R R path 2 H1(A,R) to !. To (A, ◆,,!), we associate the lattice { ! | 2 H1(A,Z) ⇢ H1(A,R)}. 77 On the other hand, to any lattice L, we associate the complex torus A = (F ⌦ C)/L. Since OF acts on (F ⌦ C) through the first component, and L is stable under this action, we may associate an endomorphism structure ◆. Further, there is a natural alternating pairing on F ⌦ C which induces a polarization on A, assuring us that it is the set of complex points of an Abelian variety, given by the formula X ((z), (w)) 7! Im(zw). Write d⌧ for the natural di↵erential on C with coordinate ⌧ , viewed as an F ⌦ C- basis for F ⌦ C. The lattice L can be recovered from A by considering the integrals nR o d⌧ | 2 H1(A,Z) . ⇡ We now have three separate ways to view Hilbert modular forms. Our next step is to give some examples of Hilbert modular forms, and use all three viewpoints to use them to learn about L-values. A.4 Damerell’s Formula Damerell’s formula relates the values of the zeta functions described above to the values of certain Hilbert modular forms, known as Eisenstein series. We give a description of the pieces that go into the construction. A.4.1 Holomorphic Eisenstein Series. We give examples of Hilbert modular forms, in terms of Definition A.3.2. First, note that we have an action of O⇥,+ on any lattice L, induced by the action of OF . Denote the set of orbits of thisF action by O⇥,+L/ , and write the orbit of ↵ 2 L by ↵O⇥,+ 2 ⇥,+L/O . F F F Definition A.4.1. Let k > 2 be an integer. The holomorphic Eisenstein series of parallel weight k is the function on the space of lattices X 1 Gk(L) = Q . (↵)k 0 6= ⇥,+↵OF 2 ⇥,+ L/OF 78 Here the sum is over nonzero orbits of the action of O⇥,+ on L. This is independent F Q of the choice of representatives ↵ since ( ku) = Nm( )ku = 1 for any totally positive unit u and any integer k. The holomorphic avatar has a simple formula on the connected component corresponding to lattices of the form OF + zOF . We write it X Q 1Gk(z) = . (A.5) ((c)z + (d))k ( c,d) The sum is over nonzero representatives ( ) of the diagonal action of O⇥,+c, d on F OF OF . i.e., we choose exactly one pair (c, d) from the elements (cu, du) as u runs over the totally positive units u 2 O⇥,+, but we exclude (0, 0) from the sum. It F converges absolutely using the fact that Im(z) > 0 for all . We conclude this section with a proposition relating the values of this Eisenstein series at certain lattices to certain values of the (d + 1)-variable L- function. Proposition A.4.2. Let K be a CM field with totally real subfield F , and fix a1, . . . , ar a set of representatives for the class group of K. Fix a CM type for K and an unramified, unitary Hecke character . Then 1 Xr L(, (k, 0)) = ⇥ ⇥ + (ai)Gk(ai).[O : O , ] K F i=1 Proof. Recall that r 1 XX (a ) L( i, (s, t)) = [O⇥ ⇥ Q . : O ,+] (↵) (↵)s(↵)t K F i=1 ↵O⇥ F The innermost sum is over representatives of nonzero orbits of the O⇥,+↵ action F on ai. We can pull out a factor of (ai) from the innermost sum, and set (↵) = 1 79 since is unramified and unitary. We are left to verify that X 1 Gk(ai) = Q . (↵)k ↵O⇥ F This is exactly the sum from Definition A.4.1. A.4.2 Maass–Shimura Operators. An important player in the story is the Maass–Shimura operator. Definition A.4.3. Let f be a Hilbert modular form of parallel weight k. Write z = ( 1 1z) = (x + iy) for the coordinate on hF , and s = (z z) = 2 . Theiy weight k Maass–Shimura operator at acts on f by the formula k @ @f ( ) = k kf s s f = k k sf + . @z @z The resulting function is a nearly holomorphic modular form of weight k + 2 and type .6 We iterate the Maass–Shimura operator by the formula j( ) = f +2 2 · · · +2 (f).k k j k k The resulting function is a nearly holomorphic modular form of weight k + 2j and type j. Using the fact that the partial derivatives @ commute for di↵erent , one @z P may show that a similar definition gives a well-defined operator j , which sends k a holomorphic Hilbert modular form of weight k to a nearly holomorphic Hilbert P P modular form of weight k + 2j and type j. Remark A.4.4. Note that we do not define what it means for a function hF ! C to be a nearly holomorphic Hilbert modular form, or what its type is. The reader may take it to be the definition that 6WeightPk+2 refers to the weight k = (k⌧ )⌧ where k⌧ = k for ⌧ 6= and k = k+2. In general, weight k + 2j refers to the weight k = (k⌧ )⌧ where k = k + 2j for all . 80 (a) a nearly holomorphic modular form of type 0 is simply a holomorphic modular form, P (b) the vector space nearly holomorphic modular forms of type j includes P the vector space of nearly holomorphic modular forms of type 0j where 0  0j  j for all , and P (c) if f is a nearly holomorphic modular form of weight k and type j⌧⌧ , then⌧ P f is a nearly holomorphic modular form of weight k+2 and type + j k ⌧ ⌧ ⌧ . When 2j < k for all , this process produces all nearly holomorphic modular P forms of weight k and type j. A.4.3 Nearly Holomorphic Eisenstein Series. Write Gk,j := P j Gk for j = (j). We find a formula for this in the following proposition.k Proposition A.4.5. Let k > 2 be an integer, and j = (j) a tuple of non-negative integers. Then a formula for Gk,j is ! Y X Q ((c) jz + (d)) Gk,j(z) = k(k + 1) j . . . (k + j 1)s Q . ((c)z + (d))k+j ( ) c,d Proof. We will prove this by induction, noting that the base case j = (0) we may compare with Equation (A.5). For the induction step, we want to verify that @f Gk,j+ = + Gk,j. Since + f = (k + 2j)sf + , we may factor out the termsk j k j @z involving variables that do not depend on z (including z, but not including s) and simply verify, using that formula, that ✓ X ◆1 X+1 ((c)z + (d)) j j s +2 = (k k j + j)s . ((c)z + (d))k+j ((c)z + (d))k+j+1 (A.6) P Write f = js 1( ( ) + ( ))k+j . For the first term of f , we have c z d k+2j✓ X ◆1 (k + 2 jj )s s . ((c)z k+j + (d)) 81 The term @f splits into two terms by the product rule. @z j @f @s X 1 @ X 1 = + js @z @z ( ( ) + ( ))k+j c z d @z ((c)z + (d))k+j Call these terms 2a and 2b respectively. Term 2a simplifies using the formula j @s = jj s +1, and combines with the (k + 2j )s f term to produce @z X X +1 1 +1 ((c)z + (d))[( + 2 ) ] jk j j s = ( jk+j)s ((c)z + (d))k+j ((c)z + (d))k+j+1 (A.7) We calculate term 2b. @ X 1 X+1 (c)(zj = ( + ) j z)s k j s (A.8) @z ((c)z + (d))k+j ((c)z + (d))k+j+1 Note that we have added an extra factor of s before the sum, in exchange for adding a factor of 1s = z z to the numerator of every term inside the sum. We now add the final result from Equation (A.7) to that from Equation (A.8) to obtain +2 f . We have arranged for the factors in front to match, as wellk j as the denominators for each term in the sum, so that we only have to add the numerators. X (c)z + d (c)(z z ) = ( + ) j+1 f k j s . k+j ((c)z + (d))k+j+1 Use the fact that (c)z + d (c)(z z) = (c)z + (d) to compare with Equation (A.6). We now have values for the Gk,j that will be used to relate their values at CM points to the values of the L-functions from the previous section. A.4.4 Damerell’s Formula. Fix representatives a1, . . . , ah for the class group of K, and let be an unramified, unitary Hecke character. We write the (d + 1)-variable L-function, recalling that the inner sum is over nonzero cosets 82 0 6= O⇥,+ 2 a O⇥,+↵ i/ .F F Xh1 X (a ) L(, ( is, t)) = [O⇥ Q . : O⇥,+] (↵)s(↵)t K F i=1 O⇥,+ ↵ F We have left our a factor of (↵) from the denominator; these are forced to be 1 by the assumptions that be unramified and unitary. In the previous section, we gave the formula for Gk,j as a holomorphic function on hF . As a function on lattices, we have ! Y X Q j(↵) Gk,j(L) = k(k + 1) ( + j . . . k j 1)s Q . (A.9) (↵)k+j O⇥,+ ↵ F Here the sum is over nonzero orbits 0 6= O⇥,+ 2 O⇥,+↵ L/ . Compare to Equation F F (A.6) to see that this is the correct formula, at least for lattices L = OF + zOF . Q We might describe y = 1 2 as the “covolume of L at ”, noting that 1 2 is theis is covolume of L. Theorem A.4.6 (Damerell’s Formula). Fix an integer k > 2, a tuple j = (j) of positive integers indexed by the real embeddings of F , and an unramified unitary Hecke character . Then we may relate special values of the (d + 1)-variable L- function with the values of various Eisenstein series at lattices corresponding to fractional ideals of K as follows: P h ( ( + ) ) = Q i=1 (ai)Gk,j(ai) L , k j, j . [O⇥ O⇥,+] ( + 1) ( + 1) j, k k . . . k j s K F Proof. Compare Equations (A.9) and (A.3). A.5 Takeaways In this section we describe some applications for Damerell’s formula as discussed in the previous two sections. Specifically, we carry over the setup from Section 3: fix a totally real field F of degree d, and a CM extension K = F (↵) with OK = OF + ↵OF . We also fix a CM type of K, i.e., a preferred extension of each 83 real embedding : F ! R to a complex embedding K ! C which we also call . Thus the set of complex embeddings of K is the set of all ’s and all ’s as runs over the real embeddings of F . A.5.1 Algebraicity of L-Values: Damerell’s Theorem. In order to talk about algebraicity, we recall the algebraic definition from Section A.3.4. For simplicity, we focus on the quadratic imaginary case. Definition A.5.1 (Definition A.3.6). Fix a base ring R. A modular form defined over R is an algebraic function f that assigns to every pair (E,!) of an elliptic curve E defined over some R-algebra S and a basis ! for ⌦1 as an S-module, E/S an element of S. A modular form of weight k is such a function which satisfies the homogeneity property that for any 2 ⇥, ( ) = kc S f E, c! c f(E,!). In particular, given a modular form f defined over a number field H, an elliptic curve E defined over H, and a generator 1! 2 ⌦ , the value f(E,!) 2 H E/H is algebraic. A similar definition, and a similar statement about algebraicity, can be made for nearly modular forms. To prove the algebraicity of certain values of L-functions, we can use our previous results relating these values to the values of Eisenstein series. Eisenstein series are in fact defined over Z, so it will be enough to show that evaluating their complex avatars at the CM points of the modular curve corresponds to evaluating their algebraic avatars at a pair consisting of an elliptic curve defined over a number field H and a generator d 2 ⌦1w C which is explicitly related to anE/ algebraic generator 2 ⌦1! by a constant dw = c · !. E/H In fact, each point of the upper half plane ⌧ 2 h corresponds to a specific complex elliptic curve E⌧ and a chosen basis dw 2 ⌦1 C. Any complex torus is anE/ elliptic curve; we let L⌧ denote the lattice Z + ⌧Z ⇢ C, and write E⌧ = C/L⌧ . 84 The projection C ! E⌧ gives an isomorphism between the cotangent bundle of E⌧ and the cotangent bundle of C. Writing w for the coordinate on C, we get a generator dw of ⌦1 C. Evaluating the holomorphic function at ⌧ 2 h correspondsE⌧/ to evaluating the algebraic function at the pair (E⌧ , dw). When ⌧ = ↵ is a CM point, the corresponding elliptic curve E↵ has complex multiplication by the quadratic imaginary field K = Q(↵). A celebrated result in explicit class field theory gives that this E↵ is defined over the Hilbert class field H of K. We also have that ⇡dw is defined over H, so that Gk (E↵, ⇡d ) = k k w ⇡ Gk(E↵, dw) = ⇡ Gk(↵) 2 H. Thus by Damerell’s Formula (Theorem A.4.6), we have P L( h j, k + j,j) =1 (ai)Gk,j(ai)/s(ai)= i k ( + 1) ( + 1)(#O⇥ 2 H(µh). ⇡ k k . . . k j )⇡k K Here we write s(ai) for the covolume of ai in F ⌦Z C. Notice that since is a character of a group of order h, it takes values in the hth roots of unity µh. This is a standard algebraicity result showing that, up to a “period” k⇡ , the value L(, k + j,j) lives in a particular number field. Above we wrote H(µh), but many authors would bound it more precisely by writing H(the values of ). In addition to algebraicity, there are integrality results, bounding the denominators of k ⇡ L(, k + j,j) 62 OH(µh). A similar construction can be carried out for a general CM field K, using Hilbert modular forms on its maximal totally real subfield and Hilbert Blumenthal Abelian varieties in place of elliptic curves. We omit it here, as it is nearly the same once we have the geometric desctiption of Hilbert modular forms from Section A.3.4. 85 A.5.2 p-adic Interpolation. One application that requires the algebraicity and integrality results described in the previous section is the construction of the p-adic L-function for a CM field. These are laid out carefully in [Kat76] and [AI19] for quadratic imaginary K (respectively for p split in K and p nonsplit in K), and in [Kat78] for a general CM field K (under the assumption that all primes of the maximal totally real subfield +K of K which divide p are split in + K/K ). The case when primes above p are nonsplit in +K/K is not settled. Very roughly, the construction goes like this. First, we relate the values of the L-function at certain inputs to the values of certain modular forms. We then modify it to write ⇣Q ⌘ Lp(, (k + j,j)) = p| 1 (p) Nm(p)k L(, (k + j,j)) =p P = [predictable constants] h [p]=1 (ai)G (ai).i k,j Here Lp is the function to be interpolated, and it is modified from L by removing the Euler factor at p. We also have [p]G , the p-depletion of the Eisenstein series k,j Gk,j. When K is quadratic imaginary, one might describe [p] G in terms of its q- k,j expansion as the form whose nth Fourier coecient is 0 whenever n is a power of p, whose nth Fourier coecient is the same as that for Gk,j when n is prime to p. Second, we prove congruences modulo np between the values Gk,0(ai) and Gk0,0(ai) whenever k ⌘ 0k (mod (p 1) n1p ). For K quadratic imaginary, this is established using a q-expansion principle in [Ser72]. Serre’s intended use is to p- adically interpolate the standard Riemann zeta function, but the result is useful here as well. P P 0 Finally, we prove congruences modulo np between j f and j f k k whenever f is a p-depleted form and each 0j ⌘ j (mod (p 1) n1p ). From here we 86 have the neccessary congruences to claim that Lp(,) is p-adically continuous as a function of the d + 1 variables. 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