First Moments of Some Hecke $L$-functions of Prime Moduli

Conjectures about discriminants of Hecke algebras at prime level

We study p-divisibility of discriminants of Hecke algebras associated to spaces of cusp forms of prime level. We make a precise conjecture about the indexes of Hecke algebras in their normalisation which implies (if true) the conjecture that there are no mod p congruences between non-conjugate newforms of weight 2 and level Gamma_0(p).

与素数相关模的二次Hecke L-函数的比率猜想

Ratios conjecture of quadratic Hecke $L$-functions of prime-related moduli

该研究在广义黎曼假设下，针对类数为1的虚二次数域中与素数相关模的二次狄利克雷及Hecke L-函数族，建立了带移位的L-函数比率猜想，并据此渐近计算了中心值一阶矩与单级密度。

On sum of Hecke eigenvalue squares over primes in very short intervals

Let $η>0$ be a fixed positive number, let $N$ be a sufficiently large number. In this paper, we study the second moment of the sum of Hecke eigenvalues over primes in short intervals (whose length is $η\log N$) on average (with some weights) over the family of weight $k$ holomorphic Hecke cusp forms. We also generalize the above result to Hecke-Maass cusp forms for $SL(2,\mathbb{Z})$ and $SL(3,\mathbb{Z}).$ By applying the Hardy-Littlewood prime 2-tuples conjecture, we calculate the exact values of the mean values.

On Hecke eigenvalues at primes of the form $[g(n)]$

In this paper, we study the average of the Fourier coefficients of a holomorphic cusp form for the full modular group at primes of the form $[g(n)]$.

On the Ramification of Hecke Algebras at Eisenstein Primes

Using the modularity technique of Wiles, we study the Hecke algebra of weight 2 and prime level N localized at the Eisenstein primes. On the way, we recover some results of Mazur ("Modular Curves and the Eisenstein Ideal") from a deformation theoretic point of view. Combining some of our results with a theorem of Merel, we obtain new information about the p-part of the class groups of Q(N^(1/p)), where p and N are prime, and N = 1 mod p.

关于海格尔-马克斯自拍形式的拉马努金素数

On Ramanujan Primes for Hecke-Maass Cusp Forms

本文研究了两个或三个不同原始海格尔-马克斯自拍形式同时满足拉马努金猜想的最小素数上界，以及至少一个形式满足猜想的素数自然密度下界。

Bounded Gaps Between Primes in Multidimensional Hecke Equidistribution Problems

Using Duke's large sieve inequality for Hecke Gr{ö}ssencharaktere and the new sieve methods of Maynard and Tao, we prove a general result on gaps between primes in the context of multidimensional Hecke equidistribution. As an application, for any fixed $0<ε<\frac{1}{2}$, we prove the existence of infinitely many bounded gaps between primes of the form $p=a^2+b^2$ such that $|a|<ε\sqrt{p}$. Furthermore, for certain diagonal curves $\mathcal{C}:ax^α+by^β=c$, we obtain infinitely many bounded gaps between the primes $p$ such that $|p+1-\#\mathcal{C}(\mathbb{F}_p)|<ε\sqrt{p}$.

关于普通素数处 Hecke 特征的反环论岩泽理论

On Anticyclotomic Iwasawa Theory of Hecke Characters at Ordinary Primes

本文研究普通素数处 Hecke 特征的岩泽理论，证明了 CM 希尔伯特模形式的反环论岩泽主猜想，并得到了 CM 阿贝尔簇的 Mordell-Weil 秩的相关结果。

Prime density results for Hecke eigenvalues of a Siegel cusp form

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.

Congruences modulo prime powers of Hecke eigenvalues in level $1$

We continue the study of strong, weak, and $dc$-weak eigenforms introduced by Chen, Kiming, and Wiese. We completely determine all systems of Hecke eigenvalues of level $1$ modulo $128$, showing there are finitely many. This extends results of Hatada and can be considered as evidence for the more general conjecture formulated by the author together with Kiming and Wiese on finiteness of systems of Hecke eigenvalues modulo prime powers at any fixed level. We also discuss the finiteness of systems of Hecke eigenvalues of level $1$ modulo $9$, reducing the question to the finiteness of a single e…

Factoring Hecke polynomials modulo a prime

Let T^{N,chi}_{p,k}(x) be the characteristic polynomial of the Hecke operator T_p acting on the space of cusp forms S_k(N,chi). We describe the factorization of T^{N,chi}_{p,k}(x) mod l as k varies, and we explicitly calculate those factorizations for N=1 and small l. These factorizations are used to deduce the irreducibility of certain T^{1,1}_{q,k}(x) from the irreducibility of T^{1,1}_{2,k}(x).

On the failure of the Gorenstein property for Hecke algebras of prime weight

In this article we report on extensive calculations concerning the Gorenstein defect for Hecke algebras of spaces of modular forms of prime weight p at maximal ideals of residue characteristic p such that the attached mod p Galois representation is unramified at p and the Frobenius at p acts by scalars. The results lead us to the ask the question whether the Gorenstein defect and the multplicity of the attached Galois representation are always equal to 2. We review the literature on the failure of the Gorenstein property and multiplicity one, discuss in some detail a very important practical i…

On sums of Hecke-Maass eigenvalues squared over primes in short intervals

We prove a uniform estimate for sums of Hecke--Maass eigenvalues squared over primes in short intervals that can be regarded as an analogue of Hoheisel's classical prime number theorem for all real analytic cusp forms. Our argument is modelled after our treatment of Linnik's least prime number theorem for arithmetic progressions (Tata LN 72) and depends on recent works in the theory of automorphic representations. We stress that constants in the present work, including those implicit, are all universal and effectively computable, although we pay no particular attention to numerical precision. …

The action of the Hecke operators on the component groups of modular Jacobian varieties

For a prime number $q\geq 5$ and a positive integer $N$ prime to $q$, Ribet proved the action of the Hecke algebra on the component group of the Jacobian variety of the modular curve of level $Nq$ at $q$ is "Eisenstein", which means the Hecke operator $T_\ell$ acts by $\ell+1$ when $\ell$ is a prime number not dividing the level. In this paper, we completely compute the action of the Hecke algebra on this component group by a careful study of supersingular points with extra automorphisms.

On Hecke Eigenvalues at Piatetski-Shapiro Primes

Let $λ(n)$ be the normalized n-th Fourier coefficient of a holomorphic cusp form for the full modular group. We show that for some constant $C > 0$ depending on the cusp form and every fixed $c$ in the range $1 < c < 8/7$, the mean value of $λ(p)$ is $\ll \exp (-C \sqrt{\log N})$ as p runs over all (Piatetski-Shapiro) primes of the form $[n^c]$ with a natural number $n \leq N$.

关于Ikeda提升的Hecke特征值

On Hecke eigenvalues of Ikeda lifts

本文证明Ikeda提升在素数处的Hecke特征值为正，并给出所有素数下的上下界，方法涉及互反多项式表达。

惯性素数处虚二次域的局部预解式、广义高斯和与反分圆Hecke $L$-值的$p$-adic赋值

The $p$-adic valuation of local resolvents, generalized Gauss sums and anticyclotomic Hecke $L$-values of imaginary quadratic fields at inert primes

本文证明了虚二次域在惯性素数处沿反分圆塔的Hecke $L$-值 $p$-adic 赋值的渐近公式，通过确定广义高斯和与局部预解式的 $p$-adic 赋值，回答了 Rubin 的问题。

Exponential sums with Dirichlet coefficients of L-functions

Improving and extending recent results of the author, we conditionally estimate exponential sums with Dirichlet coefficients of L-functions, both over all integers and over all primes in an interval. In particular, we establish new conditional results on exponential sums with Hecke eigenvalues and squares of Hecke eigenvalues over primes. We employ these estimates to improve our recent result on squares of Hecke eigenvalues at Piatetski-Shapiro primes under the Riemann Hypothesis for symmetric square L-functions for Hecke eigenforms.

Eta乘子的Hecke关系与高阶最小分部函数的同余式

Hecke Relations for Eta Multipliers and Congruences of Higher-Order Smallest Parts Functions

本研究通过作用于Eisenstein-eta商族的Hecke算子推导恒等式，得到系数素数幂模同余式，应用于解决Garvan提出的高阶最小分部函数同余问题，还建立曲柄矩、秩矩与分拆函数的素数幂模联系，强化推广了已有结果。

A note on the dimension of the largest simple Hecke submodule

For $k\ge 2$ even, let $d_{k,N}$ denote the dimension of the largest simple Hecke submodule of $S_{k}(Γ_0(N); \mathbb{Q})^\text{new}$. We show, using a simple analytic method, that $d_{k,N} \gg_k \log\log N / \log(2p)$ with $p$ the smallest prime co-prime to $N$. Previously, bounds of this quality were only known for $N$ in certain subsets of the primes. We also establish similar (and sometimes stronger) results concerning $S_{k}(Γ_0(N), χ)$, with $k \geq 2$ an integer and $χ$ an arbitrary nebentypus.

Hecke eigensystems of automorphic forms (mod $p$) of Hodge type and algebraic modular forms

We show that the systems of prime-to-$p$ Hecke eigenvalues arising from automorphic forms$\pmod p$ for a good prime $p$ associated to an algebraic group $G/\mathbb Q$ of Hodge type are the same as those arising from algebraic modular forms$\pmod p$ associated to an inner form of $G$.

一次本征形式的素数幂同余

Prime-power congruences for level-one eigenforms

本文研究权重变化时一次本征形式的素数幂同余，通过构造Hecke等变转移映射将问题简化为有限计算，并对p=2,3,5,7分别获得模2^8、3^4、5^3和7^2的全权重分类。

Construction of Hecke Characters for Three-dimensional CM Abelian Varieties

It is well-known for an elliptic curve with complex multiplication that the existence of a $\mathbb{Q}$-rational model is equivalent to its field of moduli being equal to $\mathbb{Q}$, or its endomorphism ring being the ring of integers of 9 possible fields ($\ast$). Murabayashi and Umegaki proved analogous results for abelian surfaces. For three dimensional CM abelian varieties with rational fields of moduli, Chun narrowed down to a list of 37 possible CM fields. In this paper, we show that his list is exact. By constructing certain Hecke characters that satisfy a theorem of Shimura, we prove…

Explicit results on the distribution of zeros of Hecke $L$-functions

We prove an explicit log-free zero density estimate and an explicit version of the zero-repulsion phenomenon of Deuring and Heilbronn for Hecke $L$-functions. In forthcoming work of the second author, these estimates will be used to establish explicit bounds on the least norm of a prime ideal in a congruence class group and improve upon existing explicit bounds for the least norm of a prime ideal in the Chebotarev density theorem.

Traces of Hecke operators in level 1 and Gaussian hypergeometric functions

We provide formulas for traces of p-th Hecke operators in level 1 in terms of values of finite field 2F1-hypergeometric functions, extending previous work of the author to all odd primes p, instead of only those p=1 (mod 12). We first give a general level 1 trace formula in terms of the trace of Frobenius on a family of elliptic curves, and then we draw on recent work of Lennon to produce level 1 trace formulas in terms of hypergeometric functions for all primes p >3.

关于 Hecke 特征形式非零 Fourier 系数的最大素因数

On the largest prime factor of non-zero Fourier coefficients of Hecke eigenforms

本文研究了 Ramanujan tau 函数及非 CM 规范化 Hecke 特征形式的非零 Fourier 系数的最大素因数，证明了其下界估计，改进了前人工作。

On Fourier coefficients and Hecke eigenvalues of Siegel cusp forms of degree 2

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…

某个Hecke代数商的指数

The index of a certain quotient of the Hecke algebra in its normalization

本文研究了Hecke代数在正规化下的商的指数，通过分析数域中素因子与特征形式共轭的关系，揭示了指数与素数间的关系。

关于平均最小负 Hecke 特征值

On the average least negative Hecke eigenvalue

本文研究经典 newforms 的 Hecke 特征值首次变号的平均值，区分负素数特征值与一般负特征值，并证明在 GRH 下其与最小二次非剩余均值相同。

The distribution of the eigenvalues of Hecke operators

For each prime $p$, we determine the distribution of the $p^{th}$ Fourier coefficients of the Hecke eigenforms of large weight for the full modular group. As $p\to\infty$, this distribution tends to the Sato--Tate distribution.

The first moment of twisted Hecke $L$-functions with unbounded shifts

We compute the first moment of twisted Hecke $L$-functions of prime power level going to infinity, uniformly in the conductor of the twist and in the vertical shift.

Action of Hecke operators on Siegel theta series II

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.

The Bouniakowsky conjecture and the density of polynomial roots to prime moduli

We establish a result linking the Bouniakowsky conjecture and the density of polynomial roots to prime moduli.

Primes in arithmetic progressions to large moduli, and shifted primes without large prime factors

We prove the infinitude of shifted primes $p-1$ without prime factors above $p^{0.2844}$. This refines $p^{0.2961}$ from Baker and Harman in 1998. Consequently, we obtain an improved lower bound on the the distribution of Carmichael numbers. Our main technical result is a new mean value theorem for primes in arithmetic progressions to large moduli. Namely, we estimate primes of size $x$ with quadrilinear forms of moduli up to $x^{17/32}$. This extends moduli beyond $x^{11/21}$, recently obtained by Maynard, improving $x^{29/56}$ from well-known 1986 work of Bombieri, Friedlander, and Iwaniec.

Equidistribution of Hecke points on the supersingular module

For a fixed prime p, we consider the (finite) set of supersingular elliptic curves over $\bar{\mathbb{F}}$. Hecke operators act on this set. We compute the asymptotic frequence with which a given supersingular elliptic curve visits another under this action.

Primes in arithmetic progressions to spaced moduli.III

We obtain the analog of the Bombieri-Vinogradov theorem for square moduli up to any power of x less than 1/2.

$\mathrm{GL}_2\times\mathrm{GSp}_2$ $L$-values and Hecke eigenvalue congruences

We find experimental examples of congruences of Hecke eigenvalues between automorphic representations of groups such as $\mathrm{GSp}_2(\mathbb{A})$, $\mathrm{SO}(4,3)(\mathbb{\mathbb{A}})$ and $\mathrm{SO}(5,4)(\mathbb{A})$, where the prime modulus should, for various reasons, appear in the algebraic part of a critical "tensor-product" $L$-value associated to cuspidal automorphic representations of $\mathrm{GL}_2(\mathbb{A})$ and $\mathrm{GSp}_2(\mathbb{A})$. Using special techniques for evaluating $L$-functions with few known coefficients, we compute sufficiently good approximations to detec…

Primes in arithmetic progressions to large moduli III: Uniform residue classes

We prove new mean value theorems for primes in arithmetic progressions to moduli larger than $x^{1/2}$, extending the Bombieri-Vinogradov theorem to moduli of size $x^{1/2+δ}$ which have conveniently sized divisors. The main feature of these estimates is that they are completely uniform with respect to the residue classes considered, unlike previous works on primes in arithmetic progressions to large moduli.

The critical order of certain Hecke L-functions of imaginary quadratic fields

Let $-D < -4$ denote a fundamental discriminant which is either odd or divisible by 8, so that the canonical Hecke character of $\Bbb Q(\sqrt{-D})$ exists. Let $d$ be a fundamental discriminant prime to $D$. Let $2k-1$ be an odd natural integer prime to the class number of $\Bbb Q(\sqrt{-D})$. Let $χ$ be the twist of the $(2k-1)$th power of a canonical Hecke character of $\Bbb Q(\sqrt{-D})$ by the Kronecker's symbol $n\mapsto(\frac{d}{n})$. It is proved that the order of the Hecke $L$-function $L(s,χ)$ at its central point $s=k$ is determined by its root number when $|d| \leq c(ε)D^{{1/24}-ε}$…

素数模二次Dirichlet特征和矩的界

Bounds for moments of quadratic Dirichlet character sums of prime moduli

本文在广义黎曼假设下，先对模8p的二次Dirichlet L-函数平移矩建立精确上界，再据此证明素数模二次Dirichlet特征和矩的界。

Large Sieve Inequalities for Special Characters to Prime Square Moduli

In this paper, we develop a large sieve type inequality for some special characters whose moduli are squares of primes. Our result gives non-trivial estimate in certain ranges.

On the Hecke Eigenvalues of Maass Forms

Let $ϕ$ denote a primitive Hecke-Maass cusp form for $Γ_o(N)$ with the Laplacian eigenvalue $λ_ϕ=1/4+t_ϕ^2$. In this work we show that there exists a prime $p$ such that $p\nmid N$, $|α_{p}|=|β_{p}| = 1$, and $p\ll(N(1+|t_ϕ|))^c$, where $α_{p},\;β_{p}$ are the Satake parameters of $ϕ$ at $p$, and $c$ is an absolute constant with $0<c<1$. In fact, $c$ can be taken as $0.27332$. In addition, we prove that the natural density of such primes $p$ ($p\nmid N$ and $|α_{p}|=|β_{p}| = 1$) is at least $34/35$.

半判别式为$p^2$和$4p^2$的某些三元格对应的Hecke模

Hecke modules associated to certain ternary lattices of half-discriminant $p^2$ and $4p^2$

该研究针对半判别式为$p^2$和$4p^2$的定三元格对应的Hecke模，证明其与经典模形式空间的Jacquet-Langlands型对应并给出剩余亏格猜想，通过Watson型变换实现。

Hecke grids and congruences for weakly holomorphic modular forms

Let $U(p)$ denote the Atkin operator of prime index $p$. Honda and Kaneko proved infinite families of congruences of the form $f|U(p) \equiv 0 \pmod{p}$ for weakly holomorphic modular forms of low weight and level and primes $p$ in certain residue classes, and conjectured the existence of similar congruences modulo higher powers of $p$. Partial results on some of these conjectures were proved recently by Guerzhoy. We construct infinite families of weakly holomorphic modular forms on the Fricke groups $Γ^*(N)$ for $N=1,2,3,4$ and describe explicitly the action of the Hecke algebra on these form…

Bounds on the multiplicity of the Hecke eigenvalues

Fix an integer $N$ and a prime $p\nmid N$ where $p\geq 5$. We show that the number of newforms $f$ (up to a scalar multiple) of level $N$ and even weight $k$ such that $\mathcal{T}_p(f)=0$ is bounded independently of $k$, where $\mathcal{T}_p(f)$ is the Hecke operator.

A formula for the central value of certain Hecke L-functions

Let N = 1 mod 4 be the negative of a prime, K=Q(sqrt{N}) and O_K its ring of integers. Let D be a prime ideal in O_K of prime norm congruent to 3 modulo 4. Under these assumptions, there exists Hecke characters $ψ_{\D}$ of K with conductor $(\D)$ and infinite type $(1,0)$. Their L-series L(ψ_\D,s)$ are associated to a CM elliptic curve E(N,\D) defined over the Hilbert class field of $K$. We will prove a Waldspurger-type formula for L(ψ_\D,s) of the form L(ψ_\D,1) = Ω\sum_{[\A],I} r(\D,[\A],I) m_{[\A],I}([\D]) where the sum is over class ideal representatives I of a maximal order in the quatern…

Non-vanishing of quadratic twists of modular $L$-functions of prime-related moduli

In this paper, we study central values of the family of quadratic twists of modular $L$-functions of moduli $8p$, with $p$ ranging over odd primes. Assuming the truth of the generalized Riemann hypothesis, we establish a positive proportion non-vanishing result for the corresponding $L$-values.

Primes in arithmetic progressions to large moduli II: Well-factorable estimates

We establish new mean value theorems for primes of size $x$ in arithmetic progressions to moduli as large as $x^{3/5-ε}$ when summed with suitably well-factorable weights. This extends well-known work of Bombieri, Friedlander and Iwaniec, who handled moduli of size at most $x^{4/7-ε}$. This has consequences for the level of distribution for sieve weights coming from the linear sieve.

与素数相关模数的二次狄利克雷 $L$-函数的大值

Large values of quadratic Dirichlet $L$-functions of prime-related moduli

本文在广义黎曼假设下，应用长谐振子方法，展示了与素数相关模数的二次狄利克雷 $L$-函数族在中心点处的大值。

On The Hecke Orbit Conjecture for PEL Type Shimura Varieties

The Hecke orbit conjecture asserts that every prime-to-$p$ Hecke orbit in a Shimura variety is dense in the central leaf containing it. In this paper, we prove the conjecture for certain irreducible components of Newton strata in Shimura varieties of PEL type A and C, when $p$ is an unramified prime of good reduction. Our approach generalizes Chai and Oort's method for Siegel modular varieties.

Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli

The Dirichlet characters of reduced residue systems modulo m are tabulated for moduli m <= 195. The associated L-series are tabulated for m <= 14 and small positive integer argument s accurate to 10^(-50), their first derivatives for m <= 6. Restricted summation over primes only defines Dirichlet Prime L-functions which lead to Euler products (Prime Zeta Modulo functions). Both are materialized over similar ranges of moduli and arguments. Formulas and numerical techniques are well known; the aim is to provide direct access to reference values.

On the Discrepancy of the Roots of $x^2+1$ and $x^2+2$ to Prime Moduli

In this paper, we make a conjecture (conjecture 1) related to the Bateman-Horn conjecture and proceed to study the roots of $x^2+1$ and $x^2+2$ to prime moduli, assuming the truth of the Bateman-Horn conjecture and conjecture 1 and using the Erdős-Turan-Koksma inequality.

素数幂模下二元二次型的特征和

Character sums to prime power moduli evaluated at binary quadratic forms

研究素数幂模下二元二次型的特征和估计，采用p进分析方法，补充Heath-Brown对平方自由模的估计。

Twelfth moment of Dirichlet L-functions to prime power moduli

We prove the q-aspect analogue of Heath-Brown's result on the twelfth power moment of the Riemann zeta function for Dirichlet L-functions to odd prime power moduli. Our results rely on the p-adic method of stationary phase for sums of products and complement Nunes' bound for smooth square-free moduli.