The Rankin convolution type Dirichlet series $D_{F,G}(s)$ of Siegel modular forms $F$ and $G$ of degree two, which was introduced by Kohnen and the second author, is computed numerically for various $F$ and $G$. In particular, we prove that the series $D_{F,G}(s)$, which share the same functional equation and analytic behavior with the spinor $L$-functions of eigenforms of the same weight are not linear combinations of those. In order to conduct these experiments a numerical method to compute the Petersson scalar products of Jacobi Forms is developed and discussed in detail.
In this work we study the analytic properties of the standard L-function attached to Siegel-Jacobi modular forms of higher index, generalizing previous results of Arakawa and Murase. Furthermore, we obtain algebraicity results on special L-values in the spirit of Deligne's Period Conjectures.
In this work we obtain algebraicity results on special $L$-values attached to Siegel-Jacobi modular forms. Our method relies on a generalization of the doubling method to the Jacobi group obtained in our previous work, and on introducing a notion of near holomorphy for Siegel-Jacobi modular forms. Some of our results involve also holomorphic projection, which we obtain by using Siegel-Jacobi Poincaré series of exponential type.
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…
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 $\mathrm{Sp}_4(\mathbf{F}_{\ell})$.
We prove an equidistribution statement for the Satake parameters of the local representations attached to Siegel cusp forms of degree $2$ of increasing level and weight, counted with a certain arithmetic weight. We then apply this to compute the symmetry type of a similarly weighted distribution of the low-lying zeros of $L$-functions attached to these cusp forms.
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.
Siegel modular forms of degree two attached to Hilbert modular forms
设E/Q是实二次域，pi_0是GL（2，A_E）的尖点、不可约、自守表示，具有平凡中心特征标和无穷型（2，2 n +2），其中n是非负整数。我们证明了存在一个非零Siegel仿模新形式F，它的权、水平、Hecke特征值、λ因子和L-函数都由pi_0显式确定.对于每个有理素数p，我们用pi_0的不变量将这些不变量列表。
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 $\mathrm{Re}(s)=1$.
Siegel-Shidlovskii theory of $E$-functions involves a non-vanishing proof for the determinants attached to the linear forms $D^kR(t)$, derivatives of an auxiliary function $R(t)$. Let a non-zero function $F(t)$ satisfy $m$th order linear differential equation which we shall write using the differential operator $Δ=tD$ and let $L(t)$ be any non-zero linear form of the derivatives $Δ^i F(t)$ $(i=0,...,m-1; m\ge 2)$. The determinants $\det\mathcal A_k$ attached to the linear forms $Δ^kL(t)$ have certain simple properties that allow us to give a short proof for the non-vanishing of $\det\mathcal A…
We compute the Fourier coefficients of all degree 2 Siegel-Eisenstein series of square-free level $N$ transforming with the trivial character. We then apply use these formulae to present some explicit examples of higher representation numbers attached to non-unimodular quadratic forms.
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 find nice representatives for the 0-dimensional cusps of the degree $n$ Siegel upper half-space under the action of $Γ_0(\stufe)$. To each of these we attach a Siegel Eisenstein series, and then we make explicit a result of Siegel, realizing any integral weight average Siegel theta series of arbitrary level $\stufe$ and Dirichlet character $χ_L$ modulo $\stufe$ as a linear combination of Siegel 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.
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 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 describe the effect of the differential operators defined by Boecherer-Nagaoka, Flander-Ghitza and Yamauchi on the Galois representations (conjecturally) attached to Siegel modular eigenforms.
We consider Dirichlet $L$-functions $L(s, χ)$ where $χ$ is a real, non-principal character modulo $q$. Using Pintz's refinement of Page's theorem, we prove that for $q\geq 3$ the function $L(s, χ)$ has at most one real zero $β$ with $1- 1.011/\log q < β< 1$.
For any integer $d\geq 3$ such that $-d$ is a fundamental discriminant, we show that the Dirichlet $L$-function associated with the real primitive character $χ(\cdot)=(\frac{-d}{\cdot})$ does not vanish on the positive part of the interval $[1-6.5/\sqrt{d},\ 1]. $
In this paper, we calculate the Fourier coefficients of the paramodular twist of a Siegel modular form of paramodular level $N$ by a nontrivial quadratic Dirichlet character mod $p$ for $p$ a prime not dividing $N$. As an application, these formulas can be used to verify the nonvanishing of the twist for particular examples. We also deduce that the twists of Maass forms are identically zero.
We prove that all mod $p^m$ singular forms of level $N$, degree $n+r$, and $p$-rank $r$ with $n\ge r$ are congruent mod $p^m$ to linear combinations of theta series of degree $r$ attached to quadratic forms of some level. Moreover, we prove that, the levels of theta series are of the form ``$p\mbox{-power}\times N$''. Additionally, in some cases of mod $p$ singular forms with smallest possible weight, we prove that the levels of theta series should be $p$.
We investigate Poincaré series, where we average products of terms of Fourier series of real-analytic Siegel modular forms. There are some (trivial) special cases for which the products of terms of Fourier series of elliptic modular forms and harmonic Maass forms are almost holomorphic, in which case the corresponding Poincaré series are almost holomorphic as well. In general this is not the case. The main point of this paper is the study of Siegel-Poincaré series of degree $2$ attached to products of terms of Fourier series of harmonic Siegel-Maass forms and holomorphic Siegel modular forms. …
Simple unsmoothed formulas to compute the Riemann zeta function, and Dirichlet $L$-functions to a power-full modulus, are derived by elementary means (Taylor expansions and the geometric series). The formulas enable square-root of the analytic conductor complexity, up to logarithmic loss, and have an explicit remainder term that is easy to control. The formula for zeta yields a convexity bound of the same strength as that from the Riemann-Siegel formula, up to a constant factor. Practical parameter choices are discussed.
In this article we obtain an explicit formula for certain Rankin-Selberg type Dirichlet series associated to certain Siegel cusp forms of half-integral weight. Here these Siegel cusp forms of half-integral weight are obtained from the composition of the Ikeda lift and the Eichler-Zagier-Ibukiyama correspondence. The integral weight version of the main theorem had been obtained by Katsurada and Kawamura. The result of the integral weight case is a product of $L$-function and Riemann zeta functions, while half-integral weight case is a infinite summation over negative fundamental discriminants w…
We consider $p$-adic families of Siegel eigenforms of genus $2$ and finite slope, defined as local pieces of an eigenvariety and equipped with a suitable integral structure. Under some assumptions on the residual image, we show that the image of the Galois representation associated with a family is big, in the sense that a Lie algebra attached to it contains a congruence subalgebra of non-zero level. We call Galois level of the family the largest such level. We show that it is trivial when the residual representation has full image. When the residual representation is a symmetric cube, the zer…
Let $χ$ be a real primitive character to the modulus $D$. It is proved that $$ L(1,χ)\gg (\log D)^{-2022} $$ where the implied constant is absolute and effectively computable. In the proof, the lower bound for $L(1,χ)$ is first related to the distribution of zeros of a family of Dirichlet $L$-functions in a certain region, and some results on the gaps between consecutive zeros are derived. Then, by evaluating certain discrete means of the large sieve type, a contradiction can be obtained if $L(1,χ)$ is too small.
For $m\ge 2$, let $π$ be an irreducible cuspidal automorphic representation of $GL_m(\mathbb{A}_{\mathbb{Q}})$ with unitary central character. Let $a_π(n)$ be the $n^{th}$ coefficient of the $L$-function attached to $π$. Goldfeld and Sengupta have recently obtained a bound for $\sum_{n\le x} a_π(n)$ as $x \rightarrow \infty$. For $m\ge 3$ and $π$ not a symmetric power of a $GL_2(\mathbb{A}_{\mathbb{Q}})$-cuspidal automorphic representation with not all finite primes unramified for $π$, their bound is better than all previous bounds. In this paper, we further improve the bound of Golfeld and Se…
We proof a Garrett-Böcherer decomposition of a vector valued Siegel Eisenstein series $E_{l,0}^2$ of genus 2 transforming with the Weil representation of $\text{Sp}_2(\mathbb{Z})$ on the group ring $\mathbb{C}[(L'/L)^2]$. We show that the standard zeta function associated to a vector valued common eigenform $f$ for the Weil representation can be meromorphically continued to the whole $s$-plane and that it satisfies a functional equation. The proof is based on an integral representation of this zeta function in terms of $f$ and $E_{l,0}^2$.
We construct many examples of level one Siegel modular forms in the kernel of theta operators mod $p$ by using theta series attached to positive definite quadratic forms.
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.
Assuming a conjecture on distinct zeros of Dirichlet L-functions we get asymptotic results on the average number of representations of an integer as the sum of two primes in arithmetic progression. On the other hand the existence of good error terms gives information on the the location of zeros of L-functions and possible Siegel zeros. Similar results are obtained for an integer in a congruence class expressed as the sum of two primes.
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…
We formulate and prove a version of the arithmetic Siegel-Weil formula for (zero dimensional) Shimura varieties attached to tori, equipped with some additional data. More precisely, we define a family of ``special" divisors in terms of Green functions at archimedean and non-archimedean places, and prove that their degrees coincide with the Fourier coefficients of the central derivative of an Eisenstein series. The proof relies on the usual Siegel-Weil formula to provide a direct link between both sides of the identity, and in some sense, offers a more conceptual point of view on prior results …
Let $k \geq 2$ and $N$ be positive integers and let $χ$ be a Dirichlet character modulo $N$. Let $f(z)$ be a modular form in $M_k(Γ_0(N),χ)$. Then we have a unique decomposition $f(z)=E_f(z)+S_f(z)$, where $E_f(z) \in E_k(Γ_0(N),χ)$ and $S_f(z) \in S_k(Γ_0(N),χ)$. In this paper we give an explicit formula for $E_f(z)$ in terms of Eisenstein series. Then we apply our result to certain families of eta quotients and to representations of positive integers by $2k$-ary positive definite quadratic forms in order to give an alternative version of Siegel's formula for the weighted average number of re…
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…
