A realization of the Hecke algebra on the space of period functions for Gamma_0(n)

Sums of Hecke eigenvalues over quadratic polynomials

Let f(z) = sum_n a(n) n^{(k-1)/2} e(nz) be a cusp form for Gamma_0(N), character chi and weight k geq 4. Let q(x) = x^2 + sx + t be a polynomial with integral coefficients. It is shown that sum_{n \leq X} a(q(n)) = cX + O(X^{6/7+eps}) for some constant c depending on f and q. The constant vanishes in many cases, for example if k is even. On the way a Kuznetsov formula for half-integral weight and entries having different sign is derived.

无平方因子水平下的模p新形式及其在Monsky Hecke稳定滤过上的应用

Newforms mod p in squarefree level, with applications to Monsky's Hecke-stable filtration

本文提出无平方因子水平下模p新形式的代数定义，解释Monsky描述的模2水平3模形式空间分次部分的Hecke代数，并给出特征p下仍为模形式代数自同构的重整化Atkin-Lehner对合版本。

Weight Reduction for Mod l Bianchi Modular Forms

Let K be an imaginary quadratic field with class number one and ring of integers O. We prove that mod l, a system of Hecke eigenvalues occurring in the first cohomology group of some congruence subgroup Gamma of SL(2,O) can be realized in the first cohomology with trivial coefficients after increasing the level of Gamma by l.

Bernstein-Zelevinsky derivatives, branching rules and Hecke algebras

Let $G$ be a split reductive group over a $p$-adic field $F$. Let $B$ be a Borel subgroup and $U$ the maximal unipotent subgroup of $B$. Let $ψ$ be a Whittaker character of $U$. Let $I$ be an Iwahori subgroup of $G$. We describe the Iwahori-Hecke algebra action on the Gelfand-Graev representation $(\mathrm{ind}_{U}^{G}ψ)^I$ by an explicit projective module. As a consequence, for $G=GL(n,F)$, we define and describe Bernstein-Zelevinsky derivatives of representations generated by $I$-fixed vectors in terms of the corresponding Iwahori-Hecke algebra modules. Furthermore, using Lusztig's reduction…

关于二元 $p$-进 Gamma 值的一个乘积公式

On a product formula of bivariate $p$-adic Gamma values

本文建立 $p$-进 Shintani 与全实 Hecke $L$-函数的导数公式，并据此推出二元 $p$-进 Gamma 值的乘积公式，同时说明其与 Davenport–Hasse 关系高阶解析类似物的联系。

Stable Trace Formulas and Discrete Series Representations

Let G be a reductive algebraic group over Q, and suppose that Gamma is an arithmetic subgroup of G(R) defined by congruence conditions. A basic problem in arithmetic is to determine the multiplicities of discrete series representations in L^2(Gamma \ G(R)), and in general to determine the traces of Hecke operators on these spaces. In this paper we give a conjectural formula for the traces of Hecke operators, in terms of stable distributions. It is based on a stable version of Arthur's formula for L^2-Lefschetz numbers, which is due to Kottwitz. We reduce this formula to the computation of elli…

与 Hecke 尖点特征形式关联的 L 函数的逼近

Approximation of L-functions associated to Hecke cusp eigenforms

基于 Matiyasevich 方法推导 Hecke 尖点特征形式 L 函数的逼近族，证明其收敛性并给出误差公式，成功应用于模判别式 L 函数零点定位及误差界分析。

A Class of Identities Associated with Dirichlet Series Satisfying Hecke's Functional Equation

We consider two sequences $a(n)$ and $b(n)$, $1\leq n<\infty$, generated by Dirichlet series of the forms $$\sum_{n=1}^{\infty}\frac{a(n)}{λ_n^{s}}\qquad\text{and}\qquad \sum_{n=1}^{\infty}\frac{b(n)}{μ_n^{s}},$$ satisfying a familiar functional equation involving the gamma function $Γ(s)$. A general identity is established. Appearing on one side is an infinite series involving $a(n)$ and modified Bessel functions $K_ν$, wherein on the other side is an infinite series involving $b(n)$ that is an analogue of the Hurwitz zeta function. Seven special cases, including $a(n)=τ(n)$ and $a(n)=r_k(n)$…

PEL 志村簇的动机 Hecke 代数

The motivic Hecke algebra for PEL Shimura varieties

本研究构建了作用于PEL志村簇上同调的动机Hecke代数，定义其为动机三角范畴中可构造动机自同态代数的子代数，结合内部动机理论得到带该代数作用的Chow动机，并列出适用情形。

New realizations of modular forms in Calabi-Yau threefolds arising from $ϕ^4$ theory

It has been found experimentally by Brown and Schnetz that the number of points over ${\mathbb F}_p$ of a graph hypersurface is often related to the coefficients of a modular form. In this paper I prove this relation for one example of a modular form of weight $4$ and two of weight $3$, refine the statement and suggest a method of proving it for four more of weight $4$, and use the one proved example to construct two new rigid Calabi-Yau threefolds that realize Hecke eigenforms of weight $4$ (one provably and one conjecturally).

Asymptotic densities of newforms

We define the counting function for non-analytic (Maass) newforms of Hecke congruence groups_Gamma_0(M)_. We then calculate the three main terms of this counting function and give necessary and sufficient conditions on M for this counting function to have the same shape as if it were counting eigenvalues related to a cocompact group.

Attractors of dual continued fractions

Given a Farey-type map F with full branches in the extended Hecke group Gamma_m, its dual F_# results from constructing the natural extension of F, letting time go backwards, and projecting. Although numerical simulations may suggest otherwise, we show that the domain of F_# is always tame, that is, it always contains intervals. As a main technical tool we construct, for every m=3,4,5,..., a homeomorphism M_m that simultaneously linearizes all maps with branches in Gamma_m, and show that the resulting dual linearized iterated function system satisfies the strong open set condition. We explicit…

Moduli interpretation of Eisenstein series

Let L >= 3. Using the moduli interpretation, we define certain elliptic modular forms of level Gamma(L) over any field k where 6L is invertible and k contains the Lth roots of unity. These forms generate a graded algebra R_L, which, over C, is generated by the Eisenstein series of weight 1 on Gamma(L). The main result of this article is that, when k=C, the ring R_L contains all modular forms on Gamma(L) in weights >= 2. The proof combines algebraic and analytic techniques, including the action of Hecke operators and nonvanishing of L-functions. Our results give a systematic method to produce m…

From Physics to Number Theory via Noncommutative Geometry. Part I: Quantum Statistical Mechanics of Q-lattices

This is the first installment of a paper in three parts, where we use noncommutative geometry to study the space of commensurability classes of Q-lattices and we show that the arithmetic properties of KMS states in the corresponding quantum statistical mechanical system, the theory of modular Hecke algebras, and the spectral realization of zeros of L-functions are part of a unique general picture. In this first chapter we give a complete description of the multiple phase transitions and arithmetic spontaneous symmetry breaking in dimension two. The system at zero temperature settles onto a cla…