We show that the systems of Hecke eigenvalues occurring in the spaces of Siegel modular forms (mod p) of fixed dimension g, fixed level N, and varying weight, are the same as the systems occurring in the spaces of Siegel cusp forms with the same parameters and varying weight. In particular, in the case g=1, this says that the Hecke eigensystems (mod p) coming from classical modular forms are the same as those coming from cusp forms. The proof uses restriction to the superspecial locus. We also give a comparison of cusp forms on the Satake compactification versus the toroidal compactifications of Siegel modular varieties.
For a reductive group G and a finite order Cartan-type automorphism ιof G, we construct an eigenvariety parameterizing ι-invariant cuspidal Hecke eigensystems of G. In particular, for G = Gln, we prove, any self-dual cuspidal Hecke eigensystem can be deformed in a p-adic family of self-dual cuspidal Hecke eigensystems containing a Zariski dense subset of classical points.
Let F be a totally real field and G=GSp(4)_{/F}. In this paper, we show under a weak assumption that, given a Hecke eigensystem lambda which is (p,P)-ordinary for a fixed parabolic P in G, there exists a several variable p-adic family underline{lambda} of Hecke eigensystems (all of them (p,P)-nearly ordinary) which contains lambda. The assumption is that lambda is cohomological for a regular coefficient system. If F=Q, the number of variables is three. Moreover, in this case, we construct the three variable p-adic family rho_{underline{lambda}} of Galois representations associated to underline…
In this paper we present an algorithm for computing Hecke eigensystems of Hilbert-Siegel cusp forms over real quadratic fields of narrow class number one. We give some illustrative examples using the quadratic field $\Q(\sqrt{5})$. In those examples, we identify Hilbert-Siegel eigenforms that are possible lifts from Hilbert eigenforms.
In this paper, we consider automorphic forms on $\mathrm{Sp}_4(\mathbb{A}_\mathbb{Q})$ which generate large discrete series representations of $\mathrm{Sp}_4(\mathbb{R})$ as $(\mathfrak{sp}_4(\mathbb{R}),K_\infty)$-modules. We determine the cuspidal components and the structure of the space of such automorphic forms.
Let p be an odd prime and g an integer greater or equal to 2. We prove that a finite slope Siegel cuspidal eigenform of genus g can be p-adically deformed over the g-dimensional weight space. The proof of this result relies on the construction of a family of sheaves of locally analytic overconvergent modular forms.
Let $π$ be the automorphic representation of $\GSp_4(\A)$ generated by a full level cuspidal Siegel eigenform that is not a Saito-Kurokawa lift, and $τ$ be an arbitrary cuspidal, automorphic representation of $\GL_2(\A)$. Using Furusawa's integral representation for $\GSp_4\times\GL_2$ combined with a pullback formula involving the unitary group $\GU(3,3)$, we prove that the $L$-functions $L(s,π\timesτ)$ are "nice". The converse theorem of Cogdell and Piatetski-Shapiro then implies that such representations $π$ have a functorial lifting to a cuspidal representation of $\GL_4(\A)$. Combined wit…
In this paper, we describe an algorithm for computing algebraic modular forms on compact inner forms of $\mathrm{GSp}_4$ over totally real number fields. By analogues of the Jacquet-Langlands correspondence for $\mathrm{GL}_2$, this algorithm in fact computes Hecke eigensystems of Hilbert-Siegel modular forms of genus 2. We give some examples of such eigensystems over $\Q(\sqrt{2})$.
In this article we prove several level lowering results for cuspidal automorphic representations occurring in the cohomology of the Siegel modular threefold with paramodular level structure by adapting a method of Ribet in his proof of the Serre's epsilon conjecture. The proof is purely geometric and relies on the description of the supersingular locus of certain quaternionic unitary Shimura variety and an arithmetic level raising result on this Shimura variety. The heart of the proof is a comparison of the dimension of the space of vanishing cycles on the paramodular Siegel modular threefold …
We extend the usual notion of Petersson inner product on the space of cuspidal Jacobi forms to include non-cuspidal forms as well. This is done by examining carefully the relation between certain "growth-killing" invariant differential operators on $\mathbf H_2$ and those on $\mathbf{H}_1 \times \mathbf{C}$ (here $\mathbf H_n$ denotes the Siegel upper half space of degree $n$). As applications, we can understand better the growth of Petersson norms of Fourier Jacobi coefficients of Klingen Eisenstein series, which in turn has applications to finer issues about representation numbers of quadrat…
We prove that if $F$ is a non-zero (possibly non-cuspidal) vector-valued Siegel modular form of any degree, then it has infinitely many non-zero Fourier coefficients which are indexed by half-integral matrices having odd, square-free (and thus fundamental) discriminant. The proof uses an induction argument in the setting of vector-valued modular forms. In an Appendix, as an application of a variant of our result and building upon the work of A. Pollack, we show how to obtain an unconditional proof of the functional equation of the spinor $L$-function of a holomorphic cuspidal Siegel eigenform …
We study the coherent cohomology of automorphic sheaves corresponding to Siegel modular forms $f$ of low weight on ${\rm GSp}(4)$ Shimura varieties. Inspired by the work of Prasanna--Venkatesh on singular cohomology of locally symmetric spaces, we propose a conjecture that explains all the contributions of a Hecke eigensystem to coherent cohomology in terms of the action of a motivic cohomology group. Under some technical conditions, we prove that our conjecture is equivalent to Beilinson's conjecture for the adjoint $L$-function of $f$. We also prove some unconditional results in special case…
Using the relationship between Siegel cusp forms of degree $2$ and cuspidal automorphic representations of $\mathrm{GSp}(4,\mathbb{A}_{\mathbb{Q}})$, we derive some congruences involving dimensions of spaces of Siegel cusp forms of degree $2$ and the class number of $\mathbb{Q}(\sqrt{-p})$. We also obtain some congruences between the $4$-core partition function $c_4(n)$ and dimensions of spaces of Siegel cusp forms of degree $2$.
We prove a harmonically weighted equidistribution result for the $p$-th Satake parameters of the family of automorphic cuspidal representations of $\operatorname{PGSp}(2n)$ of fixed weight $\mathtt{k}$ and prime-to-$p$ level $N\to \infty$. The main tool is a new asymptotic Petersson formula for $\operatorname{GSp}(2n)$ in the level aspect.
In this article, we distinguish Siegel cuspidal eigenforms of degree two on the full symplectic group from the signs of their Hecke eigenvalues. To establish our theorem, we obtain a result towards simultaneous sign changes of eigenvalues of two Siegel eigenforms. In the course of the proof, we also prove that the Satake $p$-parameters of two different Siegel eigenforms are distinct for a set of primes $p$ of density 1. The main ingredient to prove the latter result is the theory of Galois representations attached to Siegel eigenforms.
We give a summary of results for dimensions of spaces of cuspidal Siegel modular forms of degree 2. These results together with a list of dimensions of the irreducible representations of the finite groups GSp(4,Fp) are then used to produce bounds for dimensions of spaces of newforms with respect to principal congruence subgroups of odd square-free level.
Let $π$ be a cuspidal automorphic representation of $GL_n(\mathbb{A}_\mathbb{Q})$ which satisfies certain reasonable assumptions such as integrality of Hecke polynomials, the existence of mod $\ell$ Galois representations attached to $π$. Under Langlands functoriality of exterior $m$-th power $\wedge^m(π)$, $m=2,...,[\frac n2]$, we will construct a unique Artin representation associated to $π$. As a corollary, we obtain that such a cuspidal representation of $GL_n(\mathbb{A}_\mathbb{Q})$ satisfies the Ramanujan conjecture. We also revisit our previous work on Artin representations associated t…
In this work we use the Rankin-Selberg method to obtain results on the analytic properties of the standard $L$-function attached to vector valued Siegel modular forms. In particular we provide a detailed description of its possible poles and obtain a non-vanishing result of the twisted $L$-function beyond the usual range of absolute convergence. We remark that these results were known in this generality only in the case of scalar weight Siegel modular forms. As an interesting by-product of our work we establish the cuspidality of some theta series.
We generalize to Hilbert modular varieties of arbitrary dimension the work of W. Duke (Inventiones 1988) on the equidistribution of Heegner points and of primitive positively oriented closed geodesics in the Poincare upper half plane, subject to certain subconvexity results. We also prove vanishing results for limits of cuspidal Weyl sums associated with analogous problems for the Siegel upper half space of degree 2. In particular, these Weyl sums are associated with families of Humbert surfaces in Siegel 3-folds and of modular curves in these Humbert surfaces.
Many important analytic statements about automorphic forms, such as the analytic continuation of certain L-functions, rely on the well-known rapid decay of K-finite cusp forms on Siegel sets. We extend this here to prove a more general decay statement along sets much larger than Siegel sets, and furthermore state and prove the decay for smooth but not necessarily K-finite cusp forms. We also state a general theorem about the convergence of Rankin-Selberg integrals involving unipotent periods, closing a gap in the literature on L-functions. These properties serve as the analytic basis of a new …
We carry out some computations of vector valued Siegel modular forms of degree two, weight (k,2) and level one. Our approach is based on Satoh's description of the module of vector-valued Siegel modular forms of weight (k, 2) and an explicit description of the Hecke action on Fourier expansions. We highlight three experimental results: (1) we identify a rational eigenform in a three dimensional space of cusp forms, (2) we observe that non-cuspidal eigenforms of level one are not always rational and (3) we verify a number of cases of conjectures about congruences between classical modular forms…
I find an explicit description of modular units in terms of Siegel functions for the modular curves $X^+_{ns}(p^k)$ associated to the normalizer of a non-split Cartan subgroup of level $p^k$ where $p\not=2,3$ is a prime. The Cuspidal Divisor Class Group $\mathfrak{C}^+_{ns}(p^k)$ on $X^+_{ns}(p^k)$ is explicitly described as a module over the group ring $R = \mathbb{Z}[(\mathbb{Z}/p^k\mathbb{Z})^*/\{\pm 1\}]$. In this paper I give a formula involving generalized Bernoulli numbers $B_{2,χ}$ for $|\mathfrak{C}^+_{ns}(p^k)|$.
We give a Rankin-Selberg integral representation for the Spin (degree eight) $L$-function on $\mathrm{PGSp}_6$. The integral applies to the cuspidal automorphic representations associated to Siegel modular forms. If $π$ corresponds to a level one Siegel modular form $f$ of even weight, and if $f$ has a non-vanishing maximal Fourier coefficient (defined below), then we deduce the functional equation and finiteness of poles of the completed Spin $L$-function $Λ(π,Spin,s)$ of $π$.
Let F in S_k(Sp(2g, Z)) be a cuspidal Siegel eigenform of genus g with normalized Hecke eigenvalues mu_F(n). Suppose that the associated automorphic representation pi_F is locally tempered everywhere. For each c>0 we consider the set of primes p for which |mu_F(p)| >= c and we provide an explicit upper bound on the density of this set. In the case g=2, we also provide an explicit upper bound on the density of the set of primes p for which mu_F(p) >= c.
Let $Z$ be the quotient of the Siegel modular threefold $\mathcal{A}^{\rm sa}(2,4,8)$ which has been studied by van Geemen and Nygaard. They gave an implication that some 6-tuple $F_Z$ of theta constants which is in turn known to be a Klingen type Eisenstein series of weight 3 should be related to a holomorphic differential $(2,0)$-form on $Z$. The variety $Z$ is birationally equivalent to the tangent cone of Fermat quartic surface in the title. In this paper we first compute the L-function of two smooth resolutions of $Z$. One of these, denoted by $W$, is a kind of Igusa compactification such…
Let E/Q be a real quadratic field and pi_0 a cuspidal, irreducible, automorphic representation of GL(2,A_E) with trivial central character and infinity type (2,2n+2) for some non-negative integer n. We show that there exists a non-zero Siegel paramodular newform F with weight, level, Hecke eigenvalues, epsilon factor and L-function determined explicitly by pi_0. We tabulate these invariants in terms of those of pi_0 for every rational prime p.
Let $π$ be a cuspidal automorphic representation of $\mathrm{GSp}_4(\mathbf{A_Q})$, whose archimedean component is a holomorphic discrete series or limit of discrete series representation. If $π$ is not CAP or endoscopic, then we show that its associated $\ell$-adic Galois representations are irreducible and crystalline for $100\%$ of primes $\ell$. If, moreover, $π$ is neither an automorphic induction nor a symmetric cube lift, then we show that, for $100\%$ of primes $\ell$, the image of its mod $\ell$ Galois representation contains $\mathrm{Sp}_4(\mathbf{F}_{\ell})$.
Let $ Γ$ be a congruence subgroup of $ \mathrm{Sp}_{2n}(\mathbb Z) $. Using Poincaré series of $ K $-finite matrix coefficients of integrable discrete series representations of $ \mathrm{Sp}_{2n}(\mathbb R) $, we construct a spanning set for the space $ S_m(Γ) $ of Siegel cusp forms of weight $ m\in\mathbb Z_{>2n} $. We prove the non-vanishing of certain elements of this spanning set using Muić's integral non-vanishing criterion for Poincaré series on locally compact Hausdorff groups. Moreover, using the representation theory of $ \mathrm{Sp}_{2n}(\mathbb R) $, we study the Petersson inner pro…
In this note we study the symmetric powers of strongly modular icosahedral representations $ρ$ of ${\rm Gal} (\bar{F}/F)$, $F$ a number field, and their twisted $L$--functions. We prove that for such $ρ$, there exists a cuspidal automorphic representation $Π= Π_{\infty} \otimes Π_{f}$ of $GL_{6} (\mathbb{A}_{F})$ such that $L (s, {\rm sym}^{5} (ρ)) = L (s, Π_{f})$. One sees that ${\rm sym}^{5} (ρ)$ is twist equivalent to $ρ' \otimes {\rm sym}^{2} (ρ)$ for another modular icosahedral representation $ρ'$, and our theorem is a special case of a cuspidality criterion formulated and proved in this …
Let $p$ be an odd prime, $N$ a square-free odd positive integer prime to $p$, $π$ a $p$-ordinary cohomological irreducible cuspidal automorphic representation of $\mathrm{GSp}_4(\mathbb{A}_\mathbb{Q})$ of principal level $N$ and Iwahori level at $p$. Using a $p$-integral version of Rallis inner product formula and modularity theorems for $\mathrm{GSp}_{4/\mathbb{Q}}$ and $\mathrm{U}_{4/\mathbb{Q}}$, we establish an identity between the $p$-part of the critical value at $1$ of the degree $5$ $L$-function of $π$ twisted by the non-trivial quadratic Dirichlet character $ξ$ associated to the exten…
We define graded hyper-algebras of vector-valued Siegel modular forms, which allow us to study tensor products of the latter. We also define vector-valued Hecke operators for Siegel modular forms at all places of ${\mathbb Q}$, acting on these hyper-algebras. These definitions bridge the classical and representation theoretic approach to Siegel modular forms. Combining both the product structure and the action of Hecke operators, we prove in the case of elliptic modular forms that all cusp forms of sufficiently large weight can be obtained from products involving only two fixed Eisenstein seri…
Let pi be a cuspidal, automorphic representation of GSp(4) attached to a Siegel modular form of degree 2. We refine the method of Furusawa to obtain an integral representation for the degree-8 L-function L(s,pi x tau), where tau runs through certain cuspidal, automorphic representation of GL(2). Our calculations include the case of square-free level for the p-adic components of tau, and a wide class of archimedean types including Maass forms. As an application we obtain a special value result for L(s,pi x tau).
In this paper, we associate Galois representations to globally generic cuspidal automorphic representations on GSp(4), over a totally real field F, which are Steinberg at some finite place. This association is compatible with the local Langlands correspondence for GSp(4) studied recently in a preprint of Gan and Takeda. As a corollary, we relate the rank of the monodromy operator at p to the dimensions of the parahoric fixed spaces at p. The Galois representations are constructed by first passing to GL(4) over a CM extension, then applying the book of Harris-Taylor plus a refinement due to Tay…
Let $π$ and $π'$ be cuspidal automorphic representations of $\mathrm{GL}(n)$ and $\mathrm{GL}(n')$ with unitary central characters. We establish a new zero-free region for all $\mathrm{GL}(1)$-twists of the Rankin-Selberg $L$-function $L(s,π\timesπ')$, generalizing Siegel's celebrated work on Dirichlet $L$-functions. As an application, we prove the first unconditional Siegel-Walfisz theorem for the Dirichlet coefficients of $-L'(s,π\timesπ')/L(s,π\timesπ')$. Also, for $n\leq 8$, we extend the region of holomorphy and nonvanishing for the twisted symmetric power $L$-functions $L(s,π,\mathrm{Sym…
In connection with our previous investigation about Siegel threefolds which admit a Calabi--Yau model, we consider ball quotients which belong to the unitary group $\U(1,3)$. In this paper we determine a very particular example of a Picard modular variety of general type. Really we determine the ring of modular forms. This algebra has 25 generators, 15 modular forms $B_i$ of weight one and ten modular forms $C_j$ of weight 2. Both will appear as Borcherds products. We determine the ideal of relations. The forms $C_i$ are cuspidal. Their squares define holomorphic differential forms on the non-…
In this article we construct a $p$-adic three dimensional Eigenvariety for the group $U(2,1)(E)$, where $E$ is a quadratic imaginary field and $p$ is inert in $E$. The Eigenvariety parametrizes Hecke eigensystems on the space of overconvergent, locally analytic, cuspidal Picard modular forms of finite slope. The method generalized the one developed in Andreatta-Iovita-Pilloni by interpolating the coherent automorphic sheaves when the ordinary locus is empty. As an application of this construction, we reprove a particular case of the Bloch-Kato conjecture for some Galois characters of $E$, exte…
We consider a special theta lift $θ(f)$ from cuspidal Siegel modular forms $f$ on $\mathrm{Sp}_4$ to "modular forms" $θ(f)$ on $\mathrm{SO}(4,4)$. This lift can be considered an analogue of the Saito-Kurokawa lift, where now the image of the lift is representations of $\mathrm{SO}(4,4)$ that are quaternionic at infinity. We relate the Fourier coefficients of $θ(f)$ to those of $f$, and in particular prove that $θ(f)$ is nonzero and has algebraic Fourier coefficients if $f$ does. Restricting the $θ(f)$ to $G_2 \subseteq \mathrm{SO}(4,4)$, we obtain cuspidal modular forms on $G_2$ of arbitrarily…
We introduce the notion of Drinfeld modular forms with $A$-expansions, where instead of the usual Fourier expansion in $t^n$ ($t$ being the uniformizer at `infinity'), parametrized by $n \in \mathbb{N}$, we look at expansions in $t_a$, parametrized by $a \in A = \Fq [T]$. We construct an infinite family of eigenforms with $A$-expansions. Drinfeld modular forms with $A$-expansions have many desirable properties that allow us to explicitly compute the Hecke action. The applications of our results include: (i) various congruences between Drinfeld eigenforms; (ii) the computation of the eigensyste…
We undertake a detailed study of the lowest weight modules for the Hermitian symmetric pair (G,K), where G=Sp_4(R) and K is its maximal compact subgroup. In particular, we determine K-types and composition series, and write down explicit differential operators that navigate all the highest weight vectors of such a module starting from the unique lowest-weight vector. By rewriting these operators in classical language, we show that the automorphic forms on G that correspond to the highest weight vectors are exactly those that arise from nearly holomorphic vector-valued Siegel modular forms of d…
Integrality of Hecke eigenvalues and the growth of Hecke fields
Faculty of Education, Shinshu University（信州大学教育学部）；Faculty of Mathematics and Physics, Institute of Science and Engineering, Kanazawa University（金泽大学理工学域数理系）
We determine the number of level 1, polarized, algebraic regular, cuspidal automorphic representations of GL_n over Q of any given infinitesimal character, for essentially all n <= 8. For this, we compute the dimensions of spaces of level 1 automorphic forms for certain semisimple Z-forms of the compact groups SO_7, SO_8, SO_9 (and G_2) and determine Arthur's endoscopic partition of these spaces in all cases. We also give applications to the 121 even lattices of rank 25 and determinant 2 found by Borcherds, to level one self-dual automorphic representations of GL_n with trivial infinitesimal c…
Nearly higher Coleman theory and p-adic L-functions for $\mathrm{GSp}(4) \times \mathrm{GL}(2)$ and $\mathrm{GSp}(4) \times \mathrm{GL}(2) \times \mathrm{GL}(2)$
The aim of this paper is twofold. First, we introduce a new method for evaluating the multiplicity of a given discrete series in the space of level $1$ automorphic forms of a split classical group $G$ over $\mathbb{Z}$, and provide numerical applications in absolute rank $\leq 8$. Second, we prove a classification result for the level one cuspidal algebraic automorphic representations of ${\rm GL}_n$ over $\mathbb{Q}$ ($n$ arbitrary) whose motivic weight is $\leq 24$. In both cases, a key ingredient is a classical method based on the Weil explicit formula, which allows to disprove the existenc…
