On the numerator of the symplectic Hecke series of degree three

Simultaneous computation of Hecke operators

We present a method to compute two Hecke operators acting on a space of algebraic modular forms simultaneously based on an idea of Eichler's. We show that in certain cases this method can be used to obtain the action of the full Hecke algebra with respect to a hyperspecial subgroup and use it to compute Hecke eigenforms for compact forms of symplectic groups.

来自 Ehrhart 多项式的辛 Hecke 特征基

Symplectic Hecke eigenbases from Ehrhart polynomials

本文研究提取格多面体Ehrhart多项式系数的函数，证明其在偶数维度下为辛Hecke特征函数，并应用球函数及zeta函数理论推导出相关算术函数的解析、渐近与组合结果。

Rankin-Selberg method for Jacobi forms of integral weight and of half-integral weight on symplectic groups

In this article we show analytic properties of certain Rankin-Selberg type Dirichlet series for holomorphic Jacobi cusp forms of integral weight and of half-integral weight. The numerators of these Dirichlet series are the inner products of Fourier-Jacobi coefficients of two Jacobi cusp forms. The denominators and the range of summation of these Dirichlet series are like the ones of the Koecher-Maass series. The meromorphic continuations and functional equations of these Dirichlet series are obtained. Moreover, an identity between the Petersson norms of Jacobi forms with respect to linear isom…

Fourier Coefficients of Level 1 Hecke Eigenforms

Lehmer's 1947 conjecture on whether $τ(n)$ vanishes is still unresolved. In this context, it is natural to consider variants of Lehmer's conjecture. We determine many integers that cannot be values of $τ(n)$. For example, among the odd numbers $α$ such that $|α|<99$, we determine that $$τ(n) \notin \{-9, \pm 15, \pm 21, -25, -27, -33, \pm 35, \pm 45, \pm 49, -55, \pm 63, \pm 77, -81, \pm 91 \}.$$ Moreover, under GRH, we have that $τ(n) \neq -|α|$ and that $τ(n) \notin \{9,25,27,39,$ $75,81\}.$ We also consider the level 1 Hecke eigenforms in dimension 1 spaces of cusp forms. For example, for $…

四次 $L$-函数的比率猜想

Ratios conjecture of quartic $L$-functions of prime moduli

在广义黎曼假设下，利用多重狄利克雷级数方法，研究了四次海格尔 $L$-函数在素数模上的比率猜想，并评估了中心值的一阶矩和一级密度。

Ehrhart多项式、Hecke级数与仿射建筑

Ehrhart polynomials, Hecke series, and affine buildings

本文研究格多面体Ehrhart多项式系数与辛Hecke代数的关系，证明相关函数是Hecke特征函数，并计算其在与p-adic辛群仿射建筑顶点上的局部zeta函数，证实Vankov猜想。

Explicit Shimura's conjecture for Sp4

Shimura's conjectire (1963) concerns the rationality of the generating series for Hecke operators for the symplectic group of genus g. This conjecture wes proved by Andrianov for arbitrary genus g. For genus g=4, we explicify the rational fraction in this conjecture. Using formulas for images of double cosets, we first compute the sum of the generating series under the Satake spherical map, which is a rational fraction with polynomial coefficients. Then we recover the coefficients of this fraction as elements of the Hecke algebra using polynomial representation of basic Hecke operators under s…

三次素数模Hecke L函数的比率猜想

Ratios conjecture of cubic $L$-functions of prime moduli

本文在广义黎曼假设下，针对三次 Hecke L 函数提出比率猜想，并应用该猜想计算了中心值的第一矩和一级密度。

缠结算子与Hecke代数与参数化的p-adic简约群——经典群的情况

Opérateurs d'entrelacement et algèbres de Hecke avec paramètres d'un groupe réductif $p$-adique - le cas des groupes classiques

本文研究了p-adic简约群的缠结算子与Hecke代数的参数化问题，通过计算诱导项目生成元的终等代数，证明其同构于Hecke代数与有限群代数的半直接积。

Semisimple types for p-adic classical groups

We construct, for any symplectic, unitary or special orthogonal group over a locally compact nonarchimedean local field of odd residual characteristic, a type for each Bernstein component of the category of smooth representations, using Bushnell--Kutzko's theory of covers. Moreover, for a component corresponding to a cuspidal representation of a maximal Levi subgroup, we prove that the Hecke algebra is either abelian, or a generic Hecke algebra on an infinite dihedral group, with parameters which are, at least in principle, computable via results of Lusztig.

$L$-函数矩的多元狄利克雷级数预测：酉、辛与正交示例

Multiple Dirichlet series predictions for moments of $L$-functions: unitary, symplectic and orthogonal examples

本文用多元狄利克雷级数预测四类$L$-函数族的移位矩渐近公式，结果与CFKRS配方一致（除椭圆曲线二次扭动族需修正），并建立留数与配方项的一一对应。

Siegel modular forms of degree three and the cohomology of local systems

We give an explicit conjectural formula for the motivic Euler characteristic of an arbitrary symplectic local system on the moduli space A_3 of principally polarized abelian threefolds. The main term of the formula is a conjectural motive of Siegel modular forms of a certain type; the remaining terms admit a surprisingly simple description in terms of the motivic Euler characteristics for lower genera. The conjecture is based on extensive counts of curves of genus three and abelian threefolds over finite fields. It provides a lot of new information about vector-valued Siegel modular forms of d…

Hall-Littlewood多项式、仿射Schubert级数与格枚举

Hall-Littlewood polynomials, affine Schubert series, and lattice enumeration

本文引入Hall-Littlewood-Schubert级数，通过变量替换解决p-adic格枚举等问题，提供Hecke级数新公式，并作为Stanley-Reisner环Hilbert级数的q-模拟，推广Littlewood恒等式。