The local Tamagawa number conjecture for Hecke characters, II

Tamagawa number formula with coefficients over varieties in positive characteristic

We express the order of the pole and the leading coefficient of the L-function of a (large class of) -adic coefficients (any prime) over a quasi-projective variety over a finite field of characteristic p. This is a generalization of the result of Milne-Ramachandran with coefficients. The new key ingredient is the use of F-gauges and their equivalence in the derived category with Raynaud modules proved by Ekedahl.

An Equivariant Tamagawa Number Formula for Drinfeld Modules and Applications

We fix data $(K/F, E)$ consisting of a Galois extension $K/F$ of characteristic $p$ global fields with arbitrary abelian Galois group $G$ and a Drinfeld module $E$ defined over a certain Dedekind subring of $F$. For this data, we define a $G$-equivariant $L$-function $Θ_{K/F}^E$ and prove an equivariant Tamagawa number formula for certain Euler-completed versions of its special value $Θ_{K/F}^E(0)$. This generalizes Taelman's class number formula for the value $ζ_F^E(0)$ of the Goss zeta function $ζ_F^E$ associated to the pair $(F, E)$. Taelman's result is obtained from our result by setting $…

Tamagawa 数与函数域上椭圆曲线的挠点

Tamagawa numbers and torsion of elliptic curves over function fields

本文研究函数域上椭圆曲线 Tamagawa 数与其挠子群的关系，证明了非各向同性情形下挠阶平方整除 Tamagawa 数（除四个例外族），并给出完整刻画与猜想。

Jacobian 的 Tamagawa 数公式

Tamagawa number formula for Jacobians

本文给出离散赋值域上 Jacobian 的 Tamagawa 数乘积公式，包含四个项：单幂、环面、算术和上同调，通过推广流-割构造、利用对偶图边上的度量以及扩展 Bosch-Liu 的上同调项描述来证明。

Explicit Tamagawa numbers for certain algebraic tori over number fields

Given a number field extension $K/k$ with an intermediate field $K^+$ fixed by a central element of the corresponding Galois group of prime order $p$, we build an algebraic torus over $k$ whose rational points are elements of $K^\times$ sent to $k^\times$ via the norm map $N_{K/K^+}$. The goal is to compute the Tamagawa number of that torus explicitly via Ono's formula that expresses it as a ratio of cohomological invariants. A fairly complete and detailed description of the cohomology of the character lattice of such a torus is given when $K/k$ is Galois. Partial results including the numerat…

特征 $p$ 下可构造层的 Weil-étale 上同调与等变玉川数猜想

Weil-étale cohomology and the equivariant Tamagawa number conjecture for constructible sheaves in characteristic $p$

本文针对有限域上簇的可构造 étale 层，利用 Weil-étale 上同调证明了相关非交换 L-函数在负整数处的特殊值公式，从而给出等变玉川数猜想的几何类比并推广了若干已有结果。

Explicit exponential maps for Hecke characters at ordinary primes

In this paper we establish the explicit exponential map for the Galois representation from a Hecke character at an ordinary prime. Such explicit maps are important in verifying the Bloch-Kato conjecture for Hecke characters.

On the $p$-adic Beilinson conjecture and the equivariant Tamagawa number conjecture

Let $E/K$ be a finite Galois extension of totally real number fields with Galois group $G$. Let $p$ be an odd prime and let $r>1$ be an odd integer. The $p$-adic Beilinson conjecture relates the values at $s=r$ of $p$-adic Artin $L$-functions attached to the irreducible characters of $G$ to those of corresponding complex Artin $L$-functions. We show that this conjecture, the equivariant Iwasawa main conjecture and a conjecture of Schneider imply the `$p$-part' of the equivariant Tamagawa number conjecture for the pair $(h^0(\mathrm{Spec}(E))(r), \mathbb Z[G])$. If $r>1$ is even we obtain a sim…

First Moment of Hecke $L$-functions with quartic characters at the central point

In this paper, we study the first moment of central values of Hecke $L$-functions associated with quartic characters.

Distribution of root numbers of Hecke characters attached to some elliptic curves

In this paper, we show that an action on the set of elliptic curves with j= 1728 preserves a certain kind of symmetry on the local root number of Hecke characters attached to such elliptic curves. As a consequence, we give results on the distribution of the root numbers and their average of the aforementioned Hecke characters.

Mean values of some Hecke characters

In this paper, we evaluate a smoothed character sum involving $\sum_{m}\sum_{n}\leg {m}{n}$, with quadratic, cubic or quartic Hecke characters $\leg {m}{n}$, and the two sums over $m$ and $n$ are of comparable lengths.

Hecke eigenvalues and relations for Siegel Eisenstein series of arbitrary degree, level, and character

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.

On the Selmer group attached to a modular form and an algebraic Hecke character

We construct an Euler system of generalized Heegner cycles to bound the Selmer group associated to a modular form and an algebraic Hecke character. The main argument is based on Kolyvagin's machinery explained by Gross while the key object of the Euler system, the generalized Heegner cycles, were first considered by Bertolini, Darmon and Prasanna.

关于无立方Hecke特征的Weyl次凸界：全实情形

On Weyl's Subconvex Bound for Cube-Free Hecke characters: Totally Real Case

本文证明了全实域上无立方级Hecke特征的Weyl型次凸界，通过显式反演Motohashi公式并利用Schwartz函数与分布不变性。

Notes on the arithmetic of Hecke L-functions

This is an expository article that concerns the various related notions of algebraic idele class characters, the Groessencharaktere of Hecke, and cohomological automorphic representations of GL(1), all under the general title of algebraic Hecke characters. The first part of the article systematically lays the foundations of algebraic Hecke characters. The only pre-requisites are: basic algebraic number theory, familiarity with the adelic language, and basic sheaf theory. Observations that play a crucial role in the arithmetic of automorphic L-functions are also discussed. The second part of th…

Hecke eigenvalues and relations for degree 2 Siegel Eisenstein series

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.

Average values of quadratic Hecke character sums

We study smoothed character sums involving $\sum_{m,n} ( \frac{m}{n} )_2$, where $( \frac{m}{n} )_2$ denotes the quadratic symbol in the Gaussian field. We extend previously known results to obtain asymptotic formulas for the sums considered to larger ranges of $m$ and $n$.

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。

Non-vanishing of L-functions, the Ramanujan conjecture, and families of Hecke characters

We prove a non-vanishing result for families of $\GL_n\times\GL_n$ Rankin-Selberg $L$-functions in the critical strip, as one factor runs over twists by Hecke characters. As an application, we simplify the proof, due to Luo, Rudnick, and Sarnak, of the best known bounds towards the Generalized Ramanujan Conjecture at the infinite places for cusp forms on $\GL_n$. A key ingredient is the regularization of the units in residue classes by the use of an Arakelov ray class group.

p-adic Eisenstein-Kronecker series and non-critical values of p-adic Hecke L-functions of an Imaginary Quadratic Field when the conductor is divisible by p

We relate non-critical special values $p$-adic $L$-functions associated to algebraic Hecke characters of an imaginary quadratic number field with class number one to $p$-adic Coleman function called the $p$-adic Eisenstein-Kronecker series, when the conductors of the algebraic Hecke characters are divisible by $p$.

Sign changes of a product of Dirichlet characters and Fourier coefficients of Hecke eigenforms

Let $f\in S_k(Γ_{0}(N))$ be a normalized Hecke eigenform of even integral weight $k$ and level $N$. Let $j\ge1$ be a positive integer. We prove that for almost all primes $p$, $p\nmid N$, and for all characters $χ_{0}=\pm 1\pmod N$, the sequence $\left(χ_{0}(p^{nj})a(p^{nj})\right)_{n\in\N}$ has infinitely many sign changes. We also obtain a similar result for the sequence $\left(a(p^{j(1+2n)})\right)_{n\in\N}$ when $j$ is odd.

On Jannsen's conjecture for Hecke characters of imaginary quadratic fields

We present a collection of results on a conjecture of Jannsen about the $p$-adic realizations associated to Hecke characters over an imaginary quadratic field $K$ of class number 1. The conjecture is easy to check for Galois groups purely of local type. We prove the conjecture under a geometric regularity condition for the imaginary quadratic field $K$ at $p$, which is related to the property that a global Galois group is purely of local type. Without this regularity assumption at $p$, we present a review of the known situations in the critical case and in the non-critical case for the realiza…

通过超几何特征和得到的Hecke算子的迹

Traces of Hecke Operators via Hypergeometric Character Sums

本文提出一种一致且具几何性的方法，利用有限域上的超几何特征和，给出了算术三角群相关情形下尖点形式空间上Hecke算子迹的显式公式，适用于椭圆模曲线和Shimura曲线，并可求出特征值。

Extreme values of Hecke $L$-functions to angular characters

Let $K$ be an imaginary quadratic number field and let $L(s,ξ_{\ell})$ denote the Hecke $L$-function to an angular character $ξ_{\ell}$ with frequency $\ell$. We detect values of $\log |L(\tfrac12,ξ_{\ell})|$ with size at least $(\sqrt{2} + o_K(1))(\log X / \log \log X)^{1/2}$ along each dyadic range $X \leqslant \ell \leqslant 2X$. This result relies on the resonance method, which is applied for the first time to this family of $L$-functions, where the classification and extraction of diagonal terms depends on the geometry of the complex embedding of $K$.

t-模的等变玉河数公式及其应用

An Equivariant Tamagawa Number Formula for t-Modules and Applications

本文为$t$-模建立等变玉河数公式，推广了Drinfeld模情形，并由此证明精细Brumer-Stark猜想和精细Coates-Sinnott猜想的$t$-模类比，为后续Iwasawa理论奠定基础。

关于 Hecke 特征的 $L$-函数与反循环塔

On $L$-functions of Hecke characters and anticyclotomic towers

本文推广 Rohrlich 的结果，证明虚二次域上一类反循环扭转 Hecke 特征的 $L$-函数在中心点的零点阶数除有限例外外由根数决定，为 0 或 1。

An Eisenstein ideal for imaginary quadratic fields and the Bloch-Kato conjecture for Hecke characters

For certain algebraic Hecke characters chi of an imaginary quadratic field F we define an Eisenstein ideal in a p-adic Hecke algebra acting on cuspidal automorphic forms of GL_2/F. By finding congruences between Eisenstein cohomology classes (in the sense of G. Harder) and cuspidal classes we prove a lower bound for the index of the Eisenstein ideal in the Hecke algebra in terms of the special L-value L(0,chi). We further prove that its index is bounded from above by the order of the Selmer group of the p-adic Galois character associated to chi^{-1}. This uses the work of R. Taylor et al. on a…

Hecke L-函数中心值的第一矩的初值

First moment of central values of Hecke $L$-functions with Fixed Order Characters

本文在广义黎曼假设下，利用双重狄利克雷级数方法，研究了不同虚二次数域上Hecke L-函数中心值的平滑第一矩的渐进行为，并给出了二次族的渐进行为公式。

Twisting eigensystems of Drinfeld Hecke eigenforms by characters

We address some questions posed by Goss related to the modularity of Drinfeld modules of rank 1 defined over the field of rational functions in one variable with coefficients in a finite field. For each positive characteristic valued Dirichlet character, we introduce certain projection operators on spaces of Drinfeld modular forms with character of a given weight and type such that when applied to a Hecke eigenform return a Hecke eigenform whose eigensystem has been twisted by the given Dirichlet character. Unlike the classical case, however, the effect on Goss' $u$-expansions for these eigenf…

n阶特征的Hecke L函数的中心值分布

Distribution of central values of Hecke $L$-functions of $n$-th order characters

该研究针对n≥3的n阶Hecke特征稀疏族，证明了log|L(1/2,χ)|的单侧中心极限定理，结合相关方法与渐近公式，还得到0<k<1时2k阶矩的猜想量级上界。

分圆域由全实域的分圆 Hecke L 值生成，II

Cyclotomic fields are generated by cyclotomic Hecke {\it L}-values of totally real fields, II

本文研究全实域上单个临界分圆 Hecke L 值能否生成相应分圆 Hecke 域，将 Jun-Lee-Sun 的平凡驯顺特征标结果推广到可解全实域上的非平凡 Hecke 特征标。

${\mathbb Q}$-curves, Hecke characters and some Diophantine equations II

In the article [PV] a general procedure to study solutions of the equations $x^4-dy^2=z^p$ was presented for negative values of $d$. The purpose of the resent article is to extend our previous results to positive values of $d$. On doing so, we give a description of the extension $\mathbb{Q}(\sqrt{d},\sqrtε)/\mathbb{Q}(\sqrt{d})$ (where $ε$ is a fundamental unit) needed to prove the existence of a Hecke character over $\mathbb{Q}(\sqrt{d})$ with prescribed local conditions. We also extend some "large image" results due to Ellenberg regarding images of Galois representations coming from $\mathbb…

The second moment of Dirichlet twists of Hecke $L$-functions

Fix a Hecke cusp form $f$, and consider the $L$-function of $f$ twisted by a primitive Dirichlet character. As we range over all primitive characters of a large modulus $q$, what is the average behavior of the square of the central value of this $L$-function? Stefanicki proved an asymptotic valid only for $q$ having very few prime factors, and we extend this to almost all $q$.

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…

Selmer groups of Hecke characters and Chow groups of self products of CM elliptic curves

We bound Selmer groups attached to Grossencharacters of CM elliptic curves by the appropriate $L$-value. Our method is to use Kato's explicit reciprocity law and the Main Conjecture as proved by Rubin. These results are then used together with work of Han and fundamental results of Nekovar to bound the torsion in the image of the $p$-adic cycle class map for self products of CM elliptic curves. Our results are valid for Chow groups of arbitrary codimension for almost all $p$.

r阶特征的Hecke L-函数的二阶矩与中心值非零性

On the second moment and non-vanishing of central values of Hecke $L$-functions of $r$-th order characters

该研究针对r≥3的Hecke L-函数，利用多重Dirichlet级数理论，推导其在含2r次单位根的整体域上的扭曲矩渐近公式，证明其中心值存在正比例非零值。

关于Hecke L-函数临界值的Deligne猜想

Deligne's conjecture on the critical values of Hecke $L$-functions

本文通过将Kings-Sprang的Eisenstein-Kronecker类解释为Blasius反射动机的de Rham类，证明了任意代数Hecke特征下Deligne关于L-函数临界值的猜想，推广了Blasius仅在CM域成立的结果。

Non vanishing of Central values of modular L-functions for Hecke eigenforms of level one

Let F(z) be a newform of weight 2k and level one with a trivial character, and assume that F(z) is a non-zero eigenform of all Hecke operators. In this paper, we study nonvanishing for central values of twisted modular L-function of F.

$p$-adic Beilinson conjecture for ordinary Hecke motives associated to imaginary quadratic fields

The purpose of this article is to give an overview of the series of papers [BK1], [BK2] concerning the $p$-adic Beilinson conjecture of motives associated to Hecke characters of an imaginary quadratic field $K$, for a prime $p$ which splits in $K$.

Moments of Quadratic Hecke $L$-functions of Imaginary Quadratic Number Fields

In this paper, we study the moments of central values of Hecke $L$-functions associated with quadratic characters in $\mathbb{Q}(i)$ and $\mathbb{Q}(ω)$ with $ω= exp(2πi/3)$ and establish some quantitative non-vanishing result for these $L$-values.

$\mathbb{Q}$-curves, Hecke characters and some Diophantine equations

In this article we study the equations $x^4+dy^2=z^p$ and $x^2+dy^6=z^p$ for positive square-free values of $d$. A Frey curve over $\mathbb{Q}(\sqrt{-d})$ is attached to each primitive solution, which happens to be a $\mathbb{Q}$-curve. Our main result is the construction of a Hecke character $χ$ satisfying that the Frey elliptic curve representation twisted by $χ$ extends to $\text{Gal}_\mathbb{Q}$, therefore (by Serre's conjectures) corresponds to a newform in $S_2(n,\varepsilon)$ for explicit values of $n$ and $\varepsilon$. Following some well known results and elimination techniques (toge…

On Hecke theory for Hermitian modular 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.

Köhler关于权1 Hecke theta级数的猜想

Köhler's Conjecture on Hecke Theta Series of Weight One

通过从二次域特征诱导的二维Galois表示推导，证明了Günter Köhler关于Hecke theta级数中二次域歧义的猜想。

某些 Hecke 特征在分圆域上的 $L$-函数非零性

Nonvanishing of $L$-function of some Hecke characters on cyclotomic fields

本文证明某些 Hecke 特征在分圆域上的非零性，通过计算特征函数和 Weil 表示作用，应用于 Fermat 曲线和超椭圆曲线的 Jacobian，得到无穷多个解析秩为零的 twist。

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 $|d| \leq c(ε)D^{{1/24}-ε}$…

Simultaneous nonvanishing of Dirichlet $L$-functions and twists of Hecke-Maass L-functions

We prove that given a Hecke-Maass form $f$ for $\text{SL}(2, \mathbb{Z})$ and a sufficiently large prime $q$, there exists a primitive Dirichlet character $χ$ of conductor $q$ such that the $L$-values $L(\tfrac{1}{2}, f \otimes χ)$ and $L(\tfrac{1}{2}, χ)$ do not vanish. We expect the same method to work for any large integer $q$.

Hecke L-值 $p$-进赋值的稳定性

Stability of $p$-adic valuations of Hecke L-values

本文证明了 CM 域上代数 Hecke 特征在 $\ell$-进反循环扭曲族（$\ell\neq p$）中临界 L-值的 $p$-稳定性结果。

A note on simultaneous nonvanishing of Dirichlet $L$-functions and twists of Hecke-Maass $L$-functions

In this note, we prove that given a Hecke-Maass cusp form $f$ for $SL_2(\mathbb{Z})$ and a sufficiently large integer $q=q_1q_2$ with $q_j\asymp \sqrt{q}$ being prime numbers for $j=1,2$, there exists a primitive Dirichlet character $χ$ of conductor $q$ such that $L\left(\frac{1}{2},f\otimes χ\right)L\left(\frac{1}{2},χ\right)\neq 0$. To prove this, we establish asymptotic formulas of $L\left(\frac{1}{2},f\otimes χ\right)L\left(\frac{1}{2},χ\right)$ over the family of even primitive Dirichlet characters $χ$ of conductor $q$ for more general $q$.

Non-vanishing modulo $p$ of Hecke $L$-values over imaginary quadratic fields

Let $p$ and $q$ be two distinct odd primes. Let $K$ be an imaginary quadratic field over which $p$ and $q$ are both split. Let $Ψ$ be a Hecke character over $K$ of infinity type $(k,j)$ with $0\le-j< k$. Under certain technical hypotheses, we show that for a Zariski dense set of finite-order characters $κ$ over $K$ which factor through the $\mathbb{Z}_q^2$-extension of $K$, the $p$-adic valuation of the algebraic part of the $L$-value $L(\overline{κΨ},k+j)$ is a constant independent of $κ$. In addition, when $j=0$ and certain technical hypothesis holds, this constant is zero.

A prime-to-p version of the Grothendieck anabelian conjecture for hyperbolic curves over finite fields of characteristic p>0

In this paper, we prove a prime-to-p version of Grothendieck's anabelian conjecture for hyperbolic curves over finite fields of characteristic p>0, whose original (full profinite) version was proved by Tamagawa in the affine case and by Mochizuki in the proper case.

关于海格尔特征的德尔igne猜想

On Deligne's conjecture for Hecke characters

本文通过哈德勒和施帕赫彻尔的策略证明了海格尔L函数临界值的德尔igne猜想，核心方法为非交换几何方法。

Representations of affine Kac-Moody groups over local and global fields: a survey of some recent results

Let G be a reductive algebraic group over a local field K or a global field F. It is well know that there exists a non-trivial and interesting representation theory of the group G(K) as well as the theory of automorphic forms on the corresponding adelic group. The purpose of this paper is to give a survey of some recent constructions and results, which show that there should exist an analog of the above theories in the case when G is replaced by the corresponding affine Kac-Moody group (which is essentially built from the formal loop group G((t)) of G). Specifically we discuss the following to…

Computing groups of Hecke characters

We describe algorithms to represent and compute groups of Hecke characters. We make use of an id{è}lic point of view and obtain the whole family of such characters, including transcendental ones. We also show how to isolate the algebraic characters, which are of particular interest in number theory. This work has been implemented in Pari/GP, and we illustrate our work with a variety of explicit examples using our implementation.

On extensions of characters of affine pro-$p$ Iwahori--Hecke algebra

Let $K$ be a non-discrete non-Archimedean local field with residue characteristic $p$. Let $G$ be the group of $K$ rational points of a algebraic connected reductive group defined over $K$. In this article we compute the extensions between characters of affine pro-$p$ Iwahori--Hecke algebra $\mathcal{H}^{\text{aff}}$ over an algebraically closed field $R$ of characteristic $p$. In rank one case we deduce the relation between the blocks and $L$-packets.