Symmetric power functoriality for Hilbert modular forms

Differential operators on Hilbert modular forms

Introduction. Several authors have developed the theory of lifting from the space of modular forms of one variable to that of modular forms on the orthogonal groups attached to quadratic forms over Q (cf. [1, 4–6, 8]). Shimura [9], [10] dealt with the problem of construction of arithmetic modular forms on orthogonal groups over totally real algebraic number fields. However, he did not take up the explicit calculation of the Fourier coefficients of lifted modular forms. On the other hand, in [3], [4] we have established a correspondence Ψ k between the space S(2k−1)/2(M,χ) of modular cusp forms of half integral weight (2k − 1)/2 of level M to the space M (2) k (M,χ) of Maass forms of Siegel modular cusp forms of degree two of weight k of level M in such a way that it commutes with the actions of Hecke operators. We evaluated explicitly the Fourier coefficients of Ψ k (f) with a form f in S(2k−1)/2(M,χ), and made clear a coincidence with Shimura’s zeta functions attached to f and Andrianov’s zeta functions attached to Ψ k (f). We note that these results are closely related to Saito–Kurokawa’s conjecture concerning Siegel modular forms of degree two. Using the technique in the theory of group representation of Jacquet and Langlands, PiatetskiShapiro [7] discussed Saito–Kurokawa’s conjecture in the case of Siegel modular forms on GpSp(2, AF ) where AF is the adele ring of an arbitrary number field F . Unfortunately, it seems that his approach is difficult to use for an explicit calculation of the Fourier coefficients of the lifted forms. The first purpose of the present note is to show the existence of a correspondence ΨN ′ between Hilbert modular forms f of half integral weight with respect to the principal congruence group and Hilbert–Siegel modular forms ΨN ′(f) of degree two attached to totally real number fields. The second one

We give a new construction of p-adic overconvergent Hilbert modular forms by using Scholze’s perfectoid Shimura varieties at infinite level and the Hodge–Tate period map. The definition is analytic, closely resembling that of complex Hilbert modular forms as holomorphic functions satisfying a transformation property under congruence subgroups. As a special case, we first revisit the case of elliptic modular forms, extending recent work of Chojecki, Hansen and Johansson. We then construct sheaves of geometric Hilbert modular forms, as well as subsheaves of integral modular forms, and vary our definitions in p-adic families. We show that the resulting spaces are isomorphic as Hecke modules to earlier constructions of Andreatta, Iovita and Pilloni. Finally, we give a new direct construction of sheaves of arithmetic Hilbert modular forms, and compare this to the construction via descent from the geometric case.

Given a totally real field F and a prime integer p which is unramified in F, we construct p-adic families of overconvergent Hilbert modular forms (of non-necessarily parallel weight) as sections of, so called, overconvergent Hilbert modular sheaves. We prove that the classical Hilbert modular forms of integral weights are overconvergent in our sense. We compare our notion with Katz’s definition of p-adic Hilbert modular forms. For F = ℚ, we prove that our notion of (families of) overconvergent elliptic modular forms coincides with those of R. Coleman and V. Pilloni.

Hilbert Modular Forms Modulo pm: The Unramified Case

In 1975, Cohen constructed a kind of one-variable modular forms of half-integral weight, say [Formula: see text], whose [Formula: see text]th Fourier coefficient only occurs when [Formula: see text] is congruent to 0 or 1 modulo 4. The space of modular forms whose Fourier coefficients have the above property is called Kohnen plus space, initially introduced by Kohnen in 1980. Recently, Hiraga and Ikeda generalized the plus space to the spaces for half-integral weight Hilbert modular forms with respect to general totally real number fields. The [Formula: see text]th Fourier coefficients [Formula: see text] of a Hilbert modular form of parallel weight [Formula: see text] lying in the generalized Kohnen plus space does not vanish only if [Formula: see text] is congruent to a square modulo 4. In this paper, we use an adelic way to construct Eisenstein series of parallel half-integral weight belonging to the generalized Kohnen plus spaces and give an explicit form for their Fourier coefficients. These series give a generalization of the one introduced by Cohen. Moreover, we show that the Kohnen plus space is generated by the cusp forms and the Eisenstein series we constructed as a vector space over [Formula: see text].

In this paper, we give a method to construct a classical modular form from a Hilbert modular form. By applying this method, we can get linear formulas which relate the Fourier coefficients of the Hilbert and classical modular forms. The paper focuses on the Hilbert modular forms over real quadratic fields. We will state a construction of relations between the special values of L-functions, especially at 0, and arithmetic functions. We will also give a relation between the sum of squares functions with underlying fields $$\mathbb {Q}(\sqrt{D})$$ and $$\mathbb {Q}$$ .

Eichler investigated when there is a basis of a space of modular forms consisting of theta series attached to quaternion algebras, and treated squarefree level. Hijikata, Pizer, and Shemanske completed the solution to Eichler’s basis problem for elliptic modular forms of arbitrary level by tour-de-force trace calculations. We revisit the basis problem using the representation-theoretic perspective of the Jacquet–Langlands correspondence. Our results include: (i) a simpler proof of the solution to the basis problem for elliptic modular forms, which also allows for more flexibility in the choice of quaternion algebra; (ii) a solution to the basis problem for Hilbert modular forms; (iii) a theory of (local and global) new and old forms for quaternion algebras; and (iv) an explicit version of the Jacquet–Langlands correspondence at the level of modular forms, which is a refinement of the Hijikata–Pizer–Shemanske solution to the basis problem. Both (i) and (ii) have practical applications to computing elliptic and Hilbert modular forms. Moreover, (iii) and (iv) are desired for arithmetic applications—to illustrate, we give a simple application to Eisenstein congruences in level p 3 p^3 .

Introduction. The main object of study in this paper with respect to zero-cycles is a special class of Hilbert-Blumenthal surfaces X, which are defined over Q as smooth compactifications of quasi-projective varieties S/Q, more precisely, of coarse moduli schemes S that represent the moduli stack of polarized abelian surfaces with real multiplication by the ring of integers in a real quadratic field F = Q( √ d). We assume that d = q is a prime ≡ 1(4) and that the class number of F is 1. Then S(C), the complex points of S, can be described as H×H/ SL2(OF ), where H is the upper halfplane. In the early seventies, Hirzebruch and Zagier [HZ] defined for each integer N a curve TN on S (called Hirzebruch-Zagier cycles) and showed that their intersection numbers occur as Fourier coefficients of modular forms of level q with Nebentypes eq , the quadratic character of F/ Q. In this connection with modular forms, HirzebruchZagier cycles reveal very similar properties to Hecke correspondences on the selfproduct of the modular curve X0(q). This crucial observation of Hirzebruch and Zagier, together with Tunnell’s proof of the Tate conjecture for a product of modular curves, inspired Harder, Langlands, and Rapoport [HLR] to prove the Tate conjecture for divisors on Hilbert-Blumenthal surfaces over abelian number fields. (The proof of the Tate conjecture was then accomplished by Klingenberg [Kl] and Murty and Ramakrishnan [MR] in the general case.) From the new strategy to study torsion zero-cycles on algebraic surfaces as developed in [LS] and used in [L1], [L2] (compare also [LR]), it is clear that a crucial point is the Tate conjecture in characteristic p at good reduction primes. As one of our main results, we prove the Tate conjecture in characteristic p for primes p that split in F for a certain class of Hilbert-Blumenthal surfaces. For this we recall that to each modular cusp form f of weight 2, level q, and Nebentypes eq , that is, f ∈ S2(� 0(q), eq ) ,w e associate a Hilbert modular cusp form ˆ f ∈ S2(SL2(OF )) under the Doi-Naganuma

We will briefly review some classical results about dimension formulas for spaces of modular forms and give details about explicit computations. One of the methods we present, using the reproducing kernel, can be generalized to Hilbert modular forms and even to vector-valued Hilbert modular forms and we will present resent results on explicit dimension formulas for vector-valued Hilbert modular forms based on joint work with N.-P. Skoruppa.

Introduction. Given a totally positive quadratic form Q over a totally real number field K, one can obtain a Hilbert modular form by restricting Q to a lattice L and forming the theta series attached to L; the Fourier coefficients of the theta series are the representation numbers of Q on L. The space of Hilbert modular forms generated by all theta series attached to lattices of the same weight, level and character is invariant under a subalgebra of the Hecke algebra, hence one can (in theory) diagonalize this space of modular forms with respect to an appropriate Hecke subalgebra and infer relations on the representation numbers of the lattices. In a previous paper the author found such relations by constructing eigenforms from theta series attached to lattices of even rank which are “nice” at dyadic primes; the purpose of this paper is to extend the previous results to all lattices. We begin by proving a lemma (Lemma 1.1) which allows us to remove the restriction regarding dyadic primes. Then using our previous work we find that associated to any even rank lattice L is a family of lattices famL which is partitioned into nuclear families (which are genera when the ground field is Q), and the averaged representation numbers of these nuclear families satisfy arithmetic relations (Theorem 1.2). In §2 we define “Fourier coefficients” attached to integral ideals for a half-integral weight Hilbert modular form. Then in analogy to the case K = Q, we describe the effect of the Hecke operators on these Fourier coefficients (Theorem 2.5). In §3 we use theta series attached to odd rank lattices to construct eigenforms for the Hecke operators; the results of §2 then give us arithmetic relations on the representation numbers of the odd rank lattices. When the ground field is Q, we may assume Q(L) ⊆ Z and then these relations may be stated as

On the reduction modulo p of certain modular p-adic Galois representations

The main subject of the book is the arithmetic of zeta functions of automorphic forms. More precisely, it looks at p-adic properties of the special values of these functions. For the Riemann-zeta function this goes back to the classical Kummer congruences for Bernoulli numbers and their p-adic analytic continuation of the standard zeta functions of Siegel and modular forms and of the convolutions of Hilbert modular forms. The book is addressed to specialists in representation theory, functional analysis and algebraic geometry. Together with new results, it provides considerable background information on p-adic measures, their Mellin transforms, Siegel and Hilbert modular forms, Hecke operators acting on them, and Euler products.

We define Hilbert–Siegel modular forms and Hecke "operators" acting on them. As with Hilbert modular forms (i.e. with Siegel degree 1), these linear transformations are not linear operators until we consider a direct product of spaces of modular forms (with varying groups), modulo natural identifications we can make between certain spaces. With Hilbert–Siegel forms (i.e. with arbitrary Siegel degree) we identify several families of natural identifications between certain spaces of modular forms. We associate the Fourier coefficients of a form in our product space to even integral lattices, independent of basis and choice of coefficient rings. We then determine the action of the Hecke operators on these Fourier coefficients, paralleling the result of Hafner and Walling for Siegel modular forms (where the number field is the field of rationals).

Doi and Naganuma (see [6]) constructed a lifting map from elliptic modular forms to Hilbert modular forms in the case of a real quadratic field with narrow class number one. A Converse Theorem for Hilbert modular forms was one of their basic tools. This gives rise to the question of constructing a lifting map in the case of Jacobi forms. Here we do the first step in this direction and prove a Converse Theorem for Hilbert-Jacobi forms. Studying the connection between functions that satisfy certain transformation laws and the functional equation of their associated L-functions has value on its own and a long history. In a celebrated paper (see [9]), Hecke showed that the automorphy of a cusp form with respect to SL2(Z) is equivalent to the functional equation of its associated L-functions. That only one functional equation is needed is in a way atypical and highly depends on the fact that SL2(Z) is generated by the matrices ( 1 1 0 1 ) and ( 0 −1 1 0 ). This situation already changes if one considers cusp forms with respect to a subgroup of SL2(Z) which have a character. In this case the functional equation of twists is required (see [18]). Hecke’s work has inspired an astonishing number of people and a lot of generalizations of his “Converse Theorem” have been made, e.g. generalizations to Hilbert modular forms as mentioned above (see [6]), Siegel modular forms (see [1], [10]) or Jacobi forms (see [14],[15]). Maass showed an analogue of Hecke’s result for nonholomorphic modular forms (see [13]). He proved that these correspond to certain L-functions in quadratic fields. An outstanding generalization of a Converse Theorem for GL(n) was done by Jacquet and Langlands for n = 2 (see [11]), Jacquet, Piatetski-Shapiro, and Shalika for n = 3 (see [12]) and Cogdell and Piatetski-Shapiro for general n (see [5]). In this paper, we prove a Converse Theorem for Hilbert-Jacobi cusp forms over a totally real number field K of degree g := [K : Q] with discriminant DK and narrow class number 1. The case g = 1, i.e., Jacobi forms over Q as considered by Eichler and Zagier (see [7]), is treated in two interesting papers by Martin (see [14] and [15]). To describe our result, we consider functions φ(τ, z) from H × C into C that have a Fourier expansion with certain conditions on the Fourier coefficients (see (3.4),(3.5), and (3.6)). We show that φ is a Hilbert-Jacobi cusp form (for the definition see Section 2) if