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.
We consider the action of Hecke-type operators on Hilbert-Siegel theta series attached to lattices of even rank. We show that average Hilbert-Siegel theta series are eigenforms for these operators, and we explicitly compute the eigenvalues.
Given a Siegel theta series and a prime p not dividing the level of the theta series, we apply to the theta series the n+1 Hecke operators that generate the local Hecke algebra at p. We show that the average theta series is an eigenform and we compute the eigenvalues.
Let $F$ and $G$ be Siegel cusp forms for $\Sp_4(\Z)$ and weights $k_1, k_2$ respectively. Also let $F$ and $G$ be Hecke eigenforms lying in distinct eigen spaces. Further suppose that neither $F$ nor $G$ is a Saito-Kurokawa lift. In this article, we study simultaneous arithmetic behaviour of Hecke eigenvalues of these Hecke eigenforms.
Given two Siegel eigenforms of different weights, we determine explicit sets of Hecke eigenvalues for the two forms that must be distinct. In degree two, and under some additional conditions, we determine explicit sets of Fourier coefficients of the two forms that must be distinct.
We evaluate the action of Hecke operators on Siegel Eisenstein series of degree 2, square-free level and arbitrary character, without using knowledge of their Fourier coefficients. From this we construct a basis of simultaneous eigenforms for the full Hecke algebra, and we compute their eigenvalues. As well, we obtain Hecke relations among the Eisenstein series. Using these Hecke relations in the case that $\stufe$ is square-free and the character is trivial, we generate a basis for the space of Eisenstein series.
We evaluate the action of Hecke operators on Siegel Eisenstein series of arbitrary degree, level and character. For square-free level, we simultaneously diagonalise the space with respect to all the Hecke operators, computing the eigenvalues explicitly, and obtain a multiplicity-one result. For arbitrary level, we simultaneously diagonalise the space with respect to the Hecke operators attached to primes not dividing the level, again computing the eigenvalues explicitly.
We show that Landau-Siegel zeros for Dirichlet $L$-functions do not exist or quantum unique ergodicity for $\mathrm{GL}_2$ Hecke-Maass newforms holds with an effective rate of convergence. This follows from a more general result: Landau-Siegel zeros of Dirichlet $L$-functions repel the zeros of all other automorphic $L$-functions from the line $\mathrm{Re}(s)=1$.
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.
In this paper, we show that half of non-zero coefficients of the spinor zeta function of a Siegel cusp form of genus 2 are positive and half are negative.
Starting from a Hilbert cusp form of weight $2\kap$, we will construct a Hilbert-Siegel cusp form of weight $\kap+\frac{m}{2}$ and degree $m$ and its transfer to inner forms of symplectic groups.
In this article, the authors give a lower bound on the number of sign changes of Fourier coefficients of a non-zero degree two Siegel cusp form of even integral weight on a Hecke congruence subgroup. They also provide an explicit upper bound for the first sign change of Fourier coefficients of such Siegel cusp forms. Explicit upper bound on the first sign change of Fourier coefficients of a non-zero Siegel cusp form of even integral weight on the Siegel modular group for arbitrary genus were dealt in an earlier work of Choie, the first author and Kohnen.
We define Hilbert-Siegel modular forms and Hecke "operators" acting on them. As with Hilbert modular forms, 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 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 a basis and choice of coefficient rings. We then det…
In this paper we generalize a well-known isomorphism between the space of cusp forms of weight $k$ for a Fuchsian subgroup of the first kind $Γ\subset\mathrm{SL}_{2}(\mathbb{R})$ and the space of certain Maaß forms of weight $k$ for $Γ$ to an isomorphism between the space of Siegel cusp forms of weight $k$ for a subgroup $Γ\subset\mathrm{Sp}_{n}(\mathbb{R})$, which is commensurable with the Siegel modular group $\mathrm{Sp}_{n}(\mathbb{Z})$, and a suitable space of Siegel-Maaß forms of weight $k$ for $Γ$.
We construct a basis for the space of half-integral weight Siegel Eisenstein series of level 4N where N is odd and square-free. Then we restrict our attention to those Eisenstein series generated from elements of $Γ_0(4)$, commenting on why this restriction is necessary for our methods. We directly apply to these forms all Hecke operators attached to odd primes, and we realize the images explicitly as linear combinations of Siegel Eisenstein series. Using this information, we diagonalize the subspace of Eisenstein series generated from elements of $Γ_0(4)$, obtaining a multiplicity-one result.
We introduce an alternate set of generators for the Hecka algebra, and give an explicit formula for the action of these operators on Fourier coefficients. With this, we compute the eigenvalues of Hecke operators acting on average Siegel theta series with half-integral weight (provided the prime associated to the operators does not divide the level of the theta series). Next, we bound the eigenvalues of these operators in terms of bounds on Fourier coefficients. Then we show that the half-integral weight Kitaoka subspace is stable under all Hecke operators. Finally, we observe that an obvious i…
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 that the "naive" convolution Dirichlet series D_2(s) attached to a degree 2 Siegel Hecke cusp form F, has a pole at s=1. As an application, we write down the asymptotic formula for the partial sums of the squares of the eigenvalues of $F$ with an explicit error term. Further, as a corollary, we are able to show that the abscissa of absolute convergence of the (normalized) spinor zeta function attached to F is s = 1.
We derive an explicit upper bound for the number of systems of Hecke eigenvalues coming from Siegel modular forms (mod p) of dimension g and level N relatively prime to p. In the special case of elliptic modular forms (g=1), our result agrees with recent work of G. Herrick.
In this article, we establish quantitative results for sign changes in certain subsequences of primitive Fourier coefficients of a non-zero Siegel cusp form of arbitrary degree over congruence subgroups. As a corollary of our result for degree two Siegel cusp forms, we get sign changes of its diagonal Fourier coefficients. In the course of our proofs, we prove the non-vanishing of certain type of Fourier-Jacobi coefficients of a Siegel cusp form and all theta components of certain Jacobi cusp forms of arbitrary degree over congruence subgroups, which are also of independent interest.
In this paper, we establish a Voronoi formula for the spinor zeta function of a Siegel cusp form of genus 2. We deduce from this formula quantitative results on the number of its positive (resp. negative) coefficients in some short intervals.
The standard approach to evaluate Hecke eigenvalues of a Siegel modular eigenform F is to determine a large number of Fourier coefficients of F and then compute the Hecke action on those coefficients. We present a new method based on the numerical evaluation of F at explicit points in the upper-half space and of its image under the Hecke operators. The approach is more efficient than the standard method and has the potential for further optimization by identifying good candidates for the points of evaluation, or finding ways of lowering the truncation bound. A limitation of the algorithm is th…
We prove that if $f$ is a non zero cusp form of weight $k$ on $Γ_0(N)$ with character $χ$ such that $N/(\text{conductor }χ)$ square-free, then there exists a square-free $n\ll_ε k^{3+ε}N^{7/2+ε}$ such that $a(f,n)\neq 0$. This significantly improves the already known existential and quantitative result from previous works.
We prove multiplicity one for vector valued holomorphic Siegel modular forms of weights greater or equal to 3 and the full Siegel modular group and give a trace formula for the action of the Hecke operators T(p) in the regular cases.
We prove `polynomial in $k$' bounds on the size of the Bergman kernel for the space of holomorphic Siegel cusp forms of degree $n$ and weight $k$. When $n=1,2$ our bounds agree with the conjectural bounds on the aforementioned size, while the lower bounds match for all $n \ge 1$. For an $L^2$-normalised Siegel cusp form $F$ of degree $2$, our bound for its sup-norm is $O_ε(k^{9/4+ε})$. Further, we show that in any compact set $Ω$ (which does not depend on $k$) contained in the Siegel fundamental domain of $\mathrm{Sp}(2, \mathbb Z)$ on the Siegel upper half space, the sup-norm of $F$ is $O_Ω(k…
We prove equidistribution theorems for a family of holomorphic Siegel cusp forms of general degree in the level aspect. Our main contribution is to estimate unipotent contributions for general degree in the geometric side of Arthur's invariant trace formula in terms of Shintani zeta functions. Several applications including the vertical Sato-Tate theorem and low-lying zeros for standard $L$-functions of holomorphic Siegel cusp forms are discussed. We also show that the "non-genuine forms" which come from non-trivial endoscopic contributions by Langlands functoriality classified by Arthur are n…
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.
We compute the special values for the spinor L-function L(s,F12) in the critical strip s={12,...,19}, where F12 is the unique (up to a scalar) Siegel cusp form of degree 3 and weight 12, which was constructed by Miyawaki. These values are proportional to the product of Petersson inner products of Ramanujan's Delta by itself and the cusp form of weight 20 for SL(2,Z) by itself by a rational number and some power of Pi. We also verify this result numerically using Dokchitser's ComputeL PARI package. To our knowledge this is the first example of a spinor L-function of Siegel cusp forms of degree …
We prove that a Siegel cusp form of degree 2 for the full modular group is determined by its set of Fourier coefficients a(S) with 4 det(S) ranging over odd squarefree integers. As a key step to our result, we also prove that a classical cusp form of half-integral weight and level 4N, with N odd and squarefree, is determined by its set of Fourier coefficients a(d) with d ranging over odd squarefree integers, a result that was previously known only for Hecke eigenforms.
We investigate some key analytic properties of Fourier coefficients and Hecke eigenvalues attached to scalar-valued Siegel cusp forms $F$ of degree 2, weight $k$ and level $N$. First, assuming that $F$ is a Hecke eigenform that is not of Saito-Kurokawa type, we prove an improved bound in the $k$-aspect for the smallest prime at which its Hecke eigenvalue is negative. Secondly, we show that there are infinitely many sign changes among the Hecke eigenvalues of $F$ at primes lying in an arithmetic progression. Third, we show that there are infinitely many positive as well as infinitely many negat…
This paper is a continuation of the author's previous wotk. We supplement four results on a family of holomorphic Siegel cusp forms for $GSp_4/\mathbb{Q}$. First, we improve the result on Hecke fields. Namely, we prove that the degree of Hecke fields is unbounded on the subspace of genuine forms which do not come from functorial lift of smaller subgroups of $GSp_4$ under a conjecture in local-global compatibility and Arthur's classification for $GSp_4$. Second, we prove simultaneous vertical Sato-Tate theorem. Namely, we prove simultaneous equidistribution of Hecke eigenvalues at finitely many…
We calculate the action of some Hecke operators on spaces of modular forms spanned by the Siegel theta-series of certain genera of strongly modular lattices closely related to the Leech lattice. Their eigenforms provide explicit examples of Siegel cusp forms.
The purpose of this paper is to give an explicit dimension formula for the spaces of vector valued Siegel cusp forms of degree two with respect to a certain kind of arithmetic subgroups of the non-split Q-forms of Sp(2,R). We obtain our result by using Hashimoto and Ibukiyama's results in [HI80],[HI83] and Wakatsuki's formula in [Wak]. Our result is a generalization of formulae in [Has84,Theorem 4.1] and [Wak,Theorem 6.1].
Let $j(z)$ be the modular $j$-invariant function. Let $τ$ be an algebraic number in the complex upper half plane $\mathbb{H}$. It was proved by Schneider and Siegel that if $τ$ is not a CM point, i.e., $[\mathbb{Q}(τ):\mathbb{Q}]\neq2$, then $j(τ)$ is transcendental. Let $f$ be a harmonic weak Maass form of weight $0$ on $Γ_0(N)$. In this paper, we consider an extension of the results of Schneider and Siegel to a family of values of $f$ on Hecke orbits of $τ$. For a positive integer $m$, let $T_m$ denote the $m$-th Hecke operator. Suppose that the coefficients of the principal part of $f$ at t…
We explicitly compute the special values of the standard $L$-function $L(s, F_{12}, \mathrm{St})$ at the critical points $s\in\{-8, -6, -4, -2, 0, 1, 3, 5, 7, 9\}$, where $F_{12}$ is the unique (up to a scalar) Siegel cusp form of degree $3$ and weight $12$, which was constructed by Miyawaki. These values are proportional to the product of the Petersson norms of symmetric square of Ramanujan's $Δ$ and the cusp form of weight $20$ for ${\rm SL}_2(\mathbb{Z})$ by a rational number and some power of $π$. We use the Rankin-Selberg method and apply the Holomorphic projection to compute these values…
In this article, we prove an omega-result for the Hecke eigenvalues $λ_F(n)$ of Maass forms $F$ which are Hecke eigenforms in the space of Siegel modular forms of weight $k$, genus two for the Siegel modular group $Sp_2(\Z)$. In particular, we prove $$ λ_F(n)= Ω(n^{k-1}\text{exp} (c \frac{\sqrt{\log n}}{\log\log n})), $$ when $c>0$ is an absolute constant. This improves the earlier result $$ λ_F(n)= Ω(n^{k-1} (\frac{\sqrt{\log n}}{\log\log n})) $$ of Das and the third author. We also show that for any $n \ge 3$, one has $$ λ_F(n) \leq n^{k-1}\text{exp} \left(c_1\sqrt{\frac{\log n}{\log\log n}}…
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 prove the weak Siegel-Weil formula for the space M_2(Q) . We study the Hecke correspondence and representation numbers associated to Eichler orders, and give the explicit formula for degree of Hecke correspondence and average representation numbers over genus. This formula could recover the main results in [DuYang]. We could identify these numbers with the Fourier coefficients of Eisenstein series via Siegel-Weil formula(weak Siegel-Weil formula for M_2(Q). In the last part of this article, we reprove four square sum Theorem via Siegel-Weil formula and Kudla's matching.
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 …
Let $F$ be a vector-valued real analytic Siegel cusp eigenform of weight $(2,1)$ with the eigenvalues $-\frac 5{12}$ and 0 for the two generators of the center of the algebra consisting of all $Sp_4(\R)$-invariant differential operators on the Siegel upper half plane of degree 2. Under the assumptions (1) the validity of the transfer of automorphic representations of $GSp_4$ to $GL_4$; (2) the existence of mod $\ell$ Galois representation attached to $F$ and its lift to characteristic zero; (3) rationality of the space consisting of any such $F$; and (4) the integrality of Hecke polynomials of…
In this paper we outline the Hecke theory for Hermitian modular forms in the sense of Hel Braun for arbitrary class number of the attached imaginary-quadratic number field. The Hecke algebra turns out to be commutative. Its inert part has a structure analogous to the case of the Siegel modular group and coincides with the tensor product of its $p$-components for inert primes $p$. This leads to a characterization of the associated Siegel-Eisenstein series. The proof also involves Hecke theory for particular congruence subgroups.
Let $k > 3$ be an integer and $p$ a prime with $p > 2k-2$. Let $f$ be a newform of weight $2k-2$ and level 1 so that $f$ is ordinary at $p$ and $\barρ_{f}$ is irreducible. Under some additional hypotheses we prove that $ord_{p}(L_{alg}(k,f)) \leq ord_{p}(# S)$ where $S$ is the Pontryagin dual of the Selmer group associated to $ρ_{f} \otimes ε^{1-k}$ with $ε$ the $p$-adic cyclotomic character. We accomplish this by first constructing a congruence between the Saito-Kurokawa lift of $f$ and a non-CAP Siegel cusp form. Once this congruence is established, we use Galois representations to obtain th…
This work contains a proof of a non-trivial explicit quantitative bound in the eigenvalue aspect for the sup-norm of a SL(3,Z) Hecke-Maass cusp form restricted to a compact set.
In this paper, we study the Drinfeld cusp forms for $Γ_1(T)$ and $Γ(T)$ using Teitelbaum's interpretation as harmonic cocycles. We obtain explicit eigenvalues of Hecke operators associated to degree one prime ideals acting on the cusp forms for $Γ_1(T)$ of small weights and conclude that these Hecke operators are simultaneously diagonalizable. We also show that the Hecke operators are not diagonalizable in general for $Γ_1(T)$ of large weights, and not for $Γ(T)$ even of small weights. The Hecke eigenvalues on cusp forms for $Γ(T)$ with small weights are determined and the eigenspaces characte…
Given a self-dual cuspidal automorphic representation for GL(2) over a number field, we establish the existence of an infinite number of Hecke eigenvalues that are greater than an explicit positive constant, and an infinite number of Hecke eigenvalues that are less than an explicit negative constant. This provides an answer to a question of Serre. We also consider analogous problems for cuspidal automorphic representations that are not self-dual.
We compute the intertwining relation between the Hecke operators and the Siegel lowering operators on Siegel modular forms of arbitrary level $N$ and character $χ$ by using formulas for the action of the Hecke operators on Fourier expansions. Using an explicit description of the Satake compactification of $Γ_0^{(n)}(N) \backslash \mathfrak{H}_n$ when $N$ is squarefree we extend this to give intertwining relations for each cusp. As an application we give formulas for the action of Hecke operators on the space of Klingen--Eisenstein series of squarefree level $N$, for primes $p \nmid N$.
We address the problem of the determination of the images of the Galois representations attached to genus 2 Siegel cusp forms of level 1 having multiplicity one. These representations are symplectic. We prove that the images are as large as possible for almost every prime, if the Siegel cusp form is not a Maass spezialform and verifies two easy to check conditions.
We introduce a new trace formula of Kuznetsov type involving the central standard L-values and the Whittaker periods of cuspidal automorphic representations of PGL_n(Q) which are spherical at the archimedean place. As an application, we show a simultaneous non-vanishing of standard L-values at n-1 points on the critical strip for infinitely many Hecke-Maass cuspidal newforms.
Hecke eigenvalues of classical modular forms often encode a wealth of arithmetic data. The Satake $p$-parameters of a Siegel modular form play a role analogous to the one played by Hecke eigenvalues in the characterization of classical modular forms. In this paper we present an algorithm by which we can compute the Satake $p$-parameters of a Siegel modular form $F$ if we are given the Hecke eigenvalues of $F$ with respect to the generators of the local Hecke algebra. We compute explicit examples and relate our computations to the Ramanujan-Petersson Conjecture.
Let $k\geq 2$ and $n\geq 1$ be any integers. In this paper, we prove that all Hecke operators act trivially on the space of ordinary Drinfeld cuspforms of level $Γ_1(t^n)$ and weight $k$.
We compute the space $S_2(K(N))$ of weight $2$ Siegel paramodular cusp forms of squarefree level $N<300$. In conformance with the paramodular conjecture of A. Brumer and K. Kramer, the space is only the additive (Gritsenko) lift space of the Jacobi cusp form space $J_{2,N}^{\text{cusp}}$ except for $N=249,295$, when it further contains one nonlift newform. For these two values of $N$, the Hasse-Weil $p$-Euler factors of a relevant abelian surface match the spin $p$-Euler factors of the nonlift newform for the first two primes $p\nmid N$.
This note is concerned with the zeros of holomorphic Hecke cusp forms of large weight on the modular surface. The zeros of such forms are symmetric about three geodesic segments and we call those zeros that lie on these segments, real. Our main results give estimates for the number of real zeros as the weight goes to infinity.
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…
