Cubic structures, equivariant Euler characteristics and lattices of modular forms

Drinfeld模等变$L$函数的Euler因子及相关推广

Euler factors of equivariant $L$--functions of Drinfeld modules and beyond

该研究针对Drinfeld模等变L函数的Euler因子在0处的恒等式问题，发展通用技术证明了任意Drinfeld模下的相关恒等式，并将方法推广至高维阿贝尔t模，为等变Tamagawa数公式提供支撑。

八种不同族的三次欧拉和的评估

Evaluation of eight different families of cubic Euler sums

研究四种、五种和六种三次欧拉和，通过三种不同的分母类型计算非线性欧拉和，利用zeta值和polylogarithmic值明确表达结果。

Some evaluation of cubic Euler sums

P. Flajolet and B. Salvy \cite{FS1998} prove the famous theorem that a nonlinear Euler sum $S_{i_1i_2\cdots i_r,q}$ reduces to a combination of sums of lower orders whenever the weight $i_1+i_2+\cdots+i_r+q$ and the order $r$ are of the same parity. In this article, we develop an approach to evaluate the cubic sums $S_{1^2m,p}$ and $S_{1l_1l_2,l_3}$. By using the approach, we establish some relations involving cubic, quadratic and linear Euler sums. Specially, we prove the cubic sums $S_{1^2m,m}$ and $S_{1(2l+1)^2,2l+1}$ are reducible to zeta values, quadratic and linear sums. Moreover, we pro…

The Equivariant Tamagawa Number Conjecture for modular motives with coefficients in Hecke algebras

Under mild hypotheses on the residual representation, we prove the Equivariant Tamagawa Number Conjecture for modular motives with coefficients in universal deformation rings and Hecke algebras using a novel combination of the methods of Euler systems and Taylor-Wiles systems. We also prove the compatibility of this conjecture with specialization.

On Euler systems for adjoint Hilbert modular Galois representations

We prove the existence of Euler systems for adjoint modular Galois representations using deformations of Galois representations coming from Hilbert modular forms and relate them to $p$-adic $L$-functions under a conjectural formula for the Fitting ideals of some equivariant congruence modules for abelian base change.

The Equivariant Tamagawa Number Conjecture for modular motives with coefficients in the Hecke algebra

We propose a formulation of the Equivariant Tamagawa Number Conjecture for modular motives with coefficients in universal deformation rings and Hecke algebras; something which seems to have been heretofore missing because the complexes of Galois cohomology required were not known to be perfect. We show that the fundamental line of this conjecture satisfies the expected compatibility property at geometric points (more precisely at the points satisfying the Weight-Monodromy conjecture) and is compatible with level-lowering and level-raising. Combining these properties with the methods of Euler a…

几何Batyrev-Manin-Peyre理论对于加法群等价紧缩的渐进行为

Geometric Batyrev-Manin-Peyre for equivariant compactifications of additive groups

本文研究了给定族在光滑射影曲线上的截面模空间的渐进行为，当通用纤维为有限维向量空间的等价紧缩时，证明了在合适的Grothendieck概形环中，这些模空间的类在某种归一化下收敛到非零有效元素，为Batyrev-Manin-Peyre猜想的动机版本提供了肯定答案。

Arthur packets for $G_2$ and perverse sheaves on cubics

This paper begins the project of defining Arthur packets of all unipotent representations for the $p$-adic exceptional group $G_2$. Here we treat the most interesting case by defining and computing Arthur packets with component group $S_3$. We also show that the distributions attached to these packets are stable, subject to a hypothesis. This is done using a self-contained microlocal analysis of simple equivariant perverse sheaves on the moduli space of homogeneous cubics in two variables. In forthcoming work we will treat the remaining unipotent representations and their endoscopic classifica…

Squares in arithmetic progression over cubic fields

Euler showed that there can be no more than three integer squares in arithmetic progression. In quadratic number fields, Xarles has shown that there can be arithmetic progressions of five squares, but not of six. Here, we prove that there are no cubic number fields which contain five squares in arithmetic progression.

Shintani's zeta function is not a finite sum of Euler products

We prove that the Shintani zeta function associated to the space of binary cubic forms cannot be written as a finite sum of Euler products. Our proof also extends to several closely related Dirichlet series. This answers a question of Wright in the negative.

On the theory of higher rank Euler, Kolyvagin and Stark systems, IV: the multiplicative group

We describe a refinement of the general theory of higher rank Euler, Kolyvagin and Stark systems in the setting of the multiplicative group over arbitrary number fields. We use the refined theory to prove new results concerning the Galois structure of ideal class groups and the validity of both the equivariant Tamagawa number conjecture and of the `refined class number formula' that has been conjectured by Mazur and Rubin and by Sano. In contrast to previous work in this direction, these results require no hypotheses on the decomposition behaviour of places that are intended to rule out the ex…

Evaluations of nonlinear Euler sums of weight ten

In this paper we present a new family of identities for Euler sums and integrals of polylogarithms by using the methods of generating function and integral representations of series. Then we apply it to obtain the closed forms of all quadratic Euler sums of weight equal to ten. Furthermore, we also establish some relations between multiple zeta (star) values and nonlinear Euler sums. As applications of these relations, we give new closed form representations of several cubic Euler sums through single zeta values and linear sums. Finally, with the help of numerical computations of Mathematica o…

分圆欧拉和与多重多对数函数的围道积分及奇偶性结果

Contour Integrations and Parity Results of Cyclotomic Euler Sums and Multiple Polylogarithm Function

该研究通过级数定义扩展三角函数，用围道积分法研究分圆欧拉和及多重多对数函数的奇偶性，得到多类相关显式公式，并推导出经典及交错欧拉和奇偶性的已知公式。

Lehmer-Euler数的同余性质

Congruence properties of Lehmer-Euler numbers

本文研究Lehmer在1935年利用三次单位根定义的广义欧拉数，给出其同余性质、递推公式和显式公式，并构造新的多项式序列，得到与欧拉数及中心阶乘数相关的恒等式。

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 $…