Equalities, congruences, and quotients of zeta functions in Arithmetic Mirror Symmetry

Dynamical zeta functions and Kummer congruences

We establish a connection between the coefficients of Artin-Mazur zeta-functions and Kummer congruences. This allows to settle positively the question of the existence of a map T such that the number of fixed points of the n-th iterate of T is equal to the absolute value of the 2n-th Euler number. Also we solve a problem of Gabcke related to the coefficients of Riemann-Siegel formula.

The Lazard formal group, universal congruences and special values of zeta functions

A connection between the theory of formal groups and arithmetic number theory is established. In particular, it is shown how to construct general Almkvist--Meurman--type congruences for the universal Bernoulli polynomials that are related with the Lazard universal formal group \cite{Tempesta1}-\cite{Tempesta3}. Their role in the theory of $L$--genera for multiplicative sequences is illustrated. As an application, sequences of integer numbers are constructed. New congruences are also obtained, useful to compute special values of a new class of Riemann--Hurwitz--type zeta functions.

分圆ζ值与模L值的范数同余

Cyclotomic norm congruences for zeta values and modular $L$-values

该研究在分圆塔中建立ζ函数与模L值的同余关系，在GL₁、GL₂情形下分别推导相关同余并给出应用，为相关数论问题提供了新的研究路径。

全实域ζ函数的挠同余研究

On the Torsion Congruence for Zeta Functions of Totally Real Fields

本研究利用Shintani锥分解探究全实域ζ函数的特殊值，证明素数次域扩张下其特殊值的一类同余关系，该结果蕴含Ritter-Weiss的挠同余，对全实域非交换岩泽主猜想的证明具有关键意义。

Higher Selberg zeta functions for congruence subgroups

As a generalization of the results [KW3],we study the functional equation of the higher Selberg zeta function for congruence subgroups. To obtain the gamma factor of this function, we introduce a higher Dirichlet $L$-function. Then we determine the gamma factor explicitly in terms of the Barnes triple gamma function and the higher Dirichlet $L$-function.

Asymptotic relations for weighted finite multiple zeta values

Multiple zeta values are real numbers defined by an infinite series generalizing values of the Riemann zeta function at positive integers. Finite truncations of this series are called multiple harmonic sums and are known to have interesting arithmetic properties. When the truncation point is one less than a prime $p$, the mod $p$ values of multiple harmonic sums are called finite multiple zeta values. The present work introduces a new class of congruence for multiple harmonic sums, which we call weighted congruences. These congruences can hold modulo arbitrarily large powers of $p$. Unlike res…

A determinant formula for relative congruence zeta functions for cyclotomic function fields

Rosen M. gave a determinant formula for relative class numbers for the P-th cyclotomic function fields in the case of the monic irreducible polynomial P, which is regarded as an analogue of the classical Maillet determinant. In this paper, we will give a determinant formula for the relative congruence zeta functions for cyclotomic function fields. Our formula is regarded as a generalization of the determinant formula for the relative class number.

Selberg's zeta functions for congruence subgroups of modular groups in SL(2,R) and SL(2,C)

It is known that the Selberg zeta function for the modular group has an expression in terms of the class numbers and the fundamental units of the indefinite binary quadratic forms. In the present paper, we generalize such a expression to any congruence subgroup of the modular groups in SL(2,R) and SL(2,C).

Arithmetic expressions of Selberg's zeta functions for congruence subgroups

In Sarnak's paper, it was proved that the Selberg zeta function for SL(2,Z) is expressed in terms of the fundamental units and the class numbers of the primitive indefinite binary quadratic forms. The aim of this paper is to obtain similar arithmetic expressions of the logarithmic derivatives of the Selberg zeta functions for congruence subgroups of SL(2,Z). As applications, we study the Brun-Titchmarsh type prime geodesic theorem, the asymptotic behavior of the sum of the class number.

高余维数中的同余模与伽罗瓦上同调中的 zeta 线

Congruence modules in higher codimension and zeta lines in Galois cohomology

本文基于高余维数同余模概念，提出了检测局部环正则性的准则与形变下同余模变化的精细结果，并应用于由 Hilbert 模形式伴随动机导出的 Galois 上同调群中构造典范线。

Congruences arising from Apéry-type series for zeta values

Recently, R. Tauraso established finite $p$-analogues of famous Apéry series for $ζ(2)$ and $ζ(3).$ In this paper, we present several congruences for finite central binomial sums arising from the truncation of Apéry-type series for $ζ(4)$ and $ζ(5).$ We also prove a $p$-analogue of Zeilberger's series for $ζ(2)$ confirming a conjecture of Z. W. Sun.

The asymptotic behaviour of the number of solutions of polynomial congruences

One mentions in a lot of papers that the poles of Igusa's p-adic zeta function determine the asymptotic behavior of the number of solutions of polynomial congruences. However, no publication clarifies this connection precisely. We try to get rid of this gap.

The Riemann Zeta-Function and Hecke Congruence Subgroups. II

This is a rework of our old file, which has been left unpublished since September 1994, on an explicit spectral decomposition of the fourth power moment of the Riemann zeta-function against a weight which is the square of a Dirichlet polynomial. At this occasion we add an explicit treatment of generalized Kloosterman sums associated with arbitrary Hecke congruence subgroups (Section 15), which might have an independent interest. At the end (Section 36) of our discussion, we set out a few problems on the distribution of eigenvalues of the hyperbolic Laplacian, which appear to us to be related t…

广义欧拉数的 $p$-adic 同余及其与偶 Zeta 值的关系

$p$-adic Congruences of Generalized Euler Numbers and Relations to Even Zeta Values

本文从 $p$-adic 和分析角度研究广义欧拉数，引入同余欧拉数并建立其 $p$-adic 同余，包括解决与 Lehmer 数相关的猜想，并利用复分析方法用同余欧拉数表示偶 Zeta 值。

The summation of infinite partial fraction decomposition I: some formulae related to the Hurwitz zeta function

In this paper we establish a new summation method by expanding $\prod_{k}(1-\frac{z}{a_{k}})^{-1}$ with two approaches: the Taylor expansion and the infinite partial fraction decomposition. Here we focus on the case when $a_{k}$ is arithmetic sequence. By this summation we obtain many equalities involve Hurwitz zeta function and Gammma function.

Congruences Involving Multiple Harmonic Sums and Finite Multiple Zeta Values

Let $p$ be a prime and ${\mathfrak P}_p$ the set of positive integers which are prime to $p$. Recently, Wang and Cai proved that for every positive integer $r$ and prime $p>2$ $$ \sum_{\substack{i+j+k=p^r\\ i,j,k\in{\mathfrak P}_p}} \frac1{ijk} \equiv -2p^{r-1} B_{p-3} \pmod{p^r}, $$ where $B_{p-3}$ is the $(p-3)$-rd Bernoulli number. In this paper we prove the following analogous result: Let $n=2$ or $4$. Then for every positive integer $r\ge n/2$ and prime $p>4$ $$ \sum_{\substack{i_1+\cdots+i_n=p^r\\ i_1,\dots,i_n\in{\mathfrak P}_p}} \frac1{i_1i_2\cdots i_n} \equiv -\frac{n!}{n+1} p^{r} B_{…

Local Zeta Functions Supported on Analytic Submanifolds and Newton Polyhedra

The local zeta functions (also called Igusa's zeta functions) over p-adic fields are connected with the number of solutions of congruences and exponential sums mod p^{m}. These zeta functions are defined as integrals over open and compact subsets with respect to the Haar measure. In this paper, we introduce new integrals defined over submanifolds, or more generally, over non-degenerate complete intersection varieties, and study their connections with some arithmetical problems such as estimation of exponential sums mod p^{m}. In particular we extend Igusa's method for estimating exponential su…

Reidemeister 和 Nielsen 数：增长速率、渐进行为、动力学ζ函数和高斯同余

The Reidemeister and the Nielsen numbers: growth rate, asymptotic behavior, dynamical zeta functions and the Gauss congruences

本文从动力学角度研究Reidemeister和Nielsen数序列的增长速率、渐进行为及高斯同余，证明了相关ζ函数的有理性和序列的生长特性。

A note on arithmetic congruences

By analyzing the coefficients of the power series defining the Kubota--Leopoldt $p$-adic $L$-function associated to the non-trivial character of a real quadratic field, we prove a congruence of Ankeny--Artin--Chowla-type for prime power modulus. Additionally, we show how some classical congruences relating Bernoulli numbers and Wilson quotients fit naturally into the theory of the $p$-adic Riemann zeta function.

On the universal p-adic sigma and Weierstrass zeta functions

For primes $p>3$ we produce a new derivation of the universal $p$-adic sigma function and $p$-adic Weierstrass zeta functions of Mazur and Tate for ordinary elliptic curves by a method that highlights congruences among coefficients in Laurent expansions of elliptic functions, and works simultaneously for generalized elliptic curves defined by Weierstrass equations.

$p$-adic model theory, $p$-adic integrals, Euler products, and zeta functions of groups

We give a survey of Denef's rationality theorem on $p$-adic integrals, its uniform in $p$ versions, the relevant model theory, and a number of applications to counting subgroups of finitely generated nilpotent groups and conjugacy classes in congruence quotients of Chevalley groups over rings of integers of local fields. We then state results on analytic properties of Euler products of such $p$-adic integrals over all $p$, and an application to counting conjugacy classes in congruence quotients of certain algebraic groups over the rationals. We then briefly discuss zeta functions arising from …

A generalization of the Hasse-Witt matrix of a hypersurface

The Hasse-Witt matrix of a hypersurface in ${\mathbb P}^n$ over a finite field of characteristic $p$ gives essentially complete mod $p$ information about the zeta function of the hypersurface. But if the degree $d$ of the hypersurface is $\leq n$, the zeta function is trivial mod $p$ and the Hasse-Witt matrix is zero-by-zero. We generalize a classical formula for the Hasse-Witt matrix to obtain a matrix that gives a nontrivial congruence for the zeta function for all $d$. We also describe the differential equations satisfied by this matrix and prove that it is generically invertible.

Zeta Functions for Analytic Mappings, Log-principalization of Ideals, and Newton Polyhedra

In this paper we provide a geometric description of the possible poles of the Igusa local zeta function associated to an analytic mapping and a locally constant function, in terms of a log-principalizaton of an ideal naturally attached to the mapping. Typically our new method provides a much shorter list of possible poles compared with the previous methods. We determine the largest real part of the poles of the Igusa zeta function, and then as a corollary, we obtain an asymptotic estimation for the number of solutions of an arbitrary system of polynomial congruences in terms of the log-canonic…

Inductive construction of the p-adic zeta functions for non-commutative p-extensions of totally real fields with exponent p

We construct the p-adic zeta function for a one-dimensional (as a p-adic Lie extension) non-commutative p-extension of a totally real number field such that the finite part of its Galois group is a pgroup with exponent p. We first calculate the Whitehead groups of the Iwasawa algebra and its canonical Ore localisation by using Oliver-Taylor's theory upon integral logarithms. This calculation reduces the existence of the non-commutative p-adic zeta function to certain congruence conditions among abelian p-adic zeta pseudomeasures. Then we finally verify these congruences by using Deligne-Ribet'…

On $\infty$-adic and $v$-adic multiple zeta functions in positive characteristic

This paper pursues positive characteristic analogues of the results of Furusho, Komori, Matsumoto and Tsumura on $p$-adic multiple $L$-functions. We consider $\infty$-adic and $v$-adic multiple zeta functions concerned by Anglès, Ngo Dac and Tavares Ribeiro. Our main results in this paper consist of: (1) integral expressions of special values of $\infty$-adic multiple zeta functions, (2) integral expression of $v$-adic multiple zeta functions themselves similar to those of $p$-adic multiple $L$-functions, (3) Kummer-type congruence for the special values of $v$-adic multiple zeta functions at …

Perturbation of zeros of the Selberg zeta-function for $Γ_0(4)$

We study the asymptotic behavior of zeros of the Selberg zeta-function for the congruence subgroup $Γ_0(4)$ as a function of a one-parameter family of characters tending to the trivial character. The motivation for the study comes from observations based on numerical computations. Some of the observed phenomena lead to precise theorems that we prove and compare with the original numerical results.

Higher Apery-like numbers arising from special values of the spectral zeta function for the non-commutative harmonic oscillator

A generalization of the Apery-like numbers, which is used to describe the special values $ζ_Q(2)$ and $ζ_Q(3)$ of the spectral zeta function for the non-commutative harmonic oscillator, are introduced and studied. In fact, we give a recurrence relation for them, which shows a ladder structure among them. Further, we consider the `rational part' of the higher Apery-like numbers. We discuss several kinds of congruence relations among them, which are regarded as an analogue of the ones among Apery numbers.

An introduction to Eisenstein measures

This paper provides an introduction to Eisenstein measures, a powerful tool for constructing certain $p$-adic $L$-functions. First seen in Serre's realization of $p$-adic Dedekind zeta functions associated to totally real fields, Eisenstein measures provide a way to extend the style of congruences Kummer observed for values of the Riemann zeta function (so-called {\em Kummer congruences}) to certain other $L$-functions. In addition to tracing key developments, we discuss some challenges that arise in more general settings, concluding with some that remain open.

Period polynomial relations between formal double zeta values of odd weight

For odd $k$, we give a formula for the relations between double zeta values $ζ(r,k-r)$ with $r$ even. This formula provides a connection with the space of cusp forms on $\mathrm{SL}_2(\mathbb{Z})$. This is the odd weight analogue of a result by Gangl, Kaneko and Zagier. We also provide an answer of a question asked by Zagier about the left kernel of some matrix. Although the restricted sum statement fails in the odd weight case, we provide an asymptotical statement that replaces it. Our statement works more generally for restricted sums with any congruence condition on the first entry of the d…

Values of symmetric polynomials and a truncated analogue of the Riemann zeta function

For each positive integer n, we determine the set of symmetric functions f for which the congruence f(p/1,p/2,...,p/(p-1)) \equiv 0 mod p^n holds for all sufficiently large primes p. Our determination is conditional on a conjecture regarding the modulo p independence of Bernoulli numbers. In a recent work the author introduced a new truncated analogue of the multiple zeta function and investigated a class of relations among values of this function at positive integers. The question answered in the present work is equivalent to the determination of the relations satisfied by values of the corre…

Ramanujan Series for Epstein Zeta Functions

In the spirit of Ramanujan, we derive exponentially fast convergent series for Epstein zeta functions $ E^{\varGamma_0(N)}(z,s)$ on the Hecke congruence groups $ \varGamma_0(N),N\in\mathbb Z_{>0}$, where $z$ is an arbitrary point in the upper half-plane $ \mathfrak H$, and $s\in\mathbb Z_{>1}$. These Ramanujan series can be reformulated as integrations of modular forms, in the framework of Eichler integrals. Particular cases of these Eichler integrals recover part of the recent results reported by Wan and Zucker (arXiv:1410.7081v1).

Apéry 数的 Lucas 同余式模 $p^2$

Lucas congruences for the Apéry numbers modulo $p^2$

本文给出 Apéry 数整函数插值的泰勒系数公式，并证明对特定基 $p$ 数字的整数 $n$，$A(n)$ 满足模 $p^2$ 的 Lucas 同余式。

Apéry-like numbers arising from special values of spectral zeta functions for non-commutative harmonic oscillators

We derive an expression for the value $ζ_Q(3)$ of the spectral zeta function $ζ_Q(s)$ studied by Ichinose and Wakayama for the non-commutative harmonic oscillator defined in the work of Parmeggiani and Wakayama using a Gaussian hypergeometric function. In this study, two sequences of rational numbers, denoted $J_2(n)$ and $J_3(n)$, which can be regarded as analogues of the Apéry numbers, naturally arise and play a key role in obtaining the expressions for the values $ζ_Q(2)$ and $ζ_Q(3)$. We also show that the numbers $J_2(n)$ and $J_3(n)$ have congruence relations like those of the Apéry numb…

Approximation of the derivatives of the logarithm of the Riemann zeta-function in the critical strip

Recently, we have established the generalized Li criterion equivalent to the Riemann hypothesis, viz. demonstrated that the sums over all non-trivial Riemann function zeroes k_n,a=Sum_(/rho)(1-(1-((/rho-a)/(/rho+a-1))^n) for any real a not equal to 1/2 are non-negative if and only if the Riemann hypothesis holds true, and proved the relation k_n,a=n*(1-2a)/(n-1)!*d^n/dz^n((z-a)^(n-1)*ln(ξ(z))) taken at z=1-a. Assuming that the function /zeta(s) is non-vanishing for Re(s)>1/2+/Delta, where real 0</Delta<1/2, using this relation together with the functional equation for the /xi-function and the …

算术群 Selberg zeta 函数的普适性定理

Universality theorems of the Selberg zeta functions for arithmetic groups

本文将 Selberg zeta 函数的普适性定理从模群及其主同余子群进一步推广到模群的一般子群和由不定四元数代数导出的余紧算术群，并得到了比已有结果更宽的成立区域。

高阶 Rademacher 符号的整性

Integrality of the higher Rademacher symbols

本文证明 Duke 关于高阶 Rademacher 符号取整数值的猜想，借助 Bernoulli 数性质及 Kummer 同余的推广，阐明相关实二次域 zeta 值分母的简洁性。

带艾森斯坦级数的同余与μ不变量

Congruences with Eisenstein series and mu-invariants

本文研究带剩余可约伽罗瓦表示的Hida族中μ不变量的变化，证明其下界通常可用p进ζ函数表示且无界，在特定条件下证明下界为等式及p进L函数的性质，并证明相关Selmer群的类似结论，确立了这类模形式的主猜想。

New properties of multiple harmonic sums modulo $p$ and $p$-analogues of Leshchiner's series

In this paper we present some new identities of hypergeometric type for multiple harmonic sums whose indices are the sequences $(\{1\}^a,c,\{1\}^b),$ $(\{2\}^a,c,\{2\}^b)$ and prove a number of congruences for these sums modulo a prime $p.$ The congruences obtained allow us to find nice $p$-analogues of Leshchiner's series for zeta values and to refine a result due to M. Hoffman and J. Zhao about the set of generators of the multiple harmonic sums of weight 7 and 9 modulo $p$. Moreover, we are also able to provide a new proof of Zagier's formula for $ζ^{*}(\{2\}^a,3,\{2\}^b)$ based on a finite…

临界带中塞尔伯格zeta函数的欧拉乘积

Euler products of Selberg zeta functions in the critical strip

该研究扩展了模群同余子群对应塞尔伯格zeta函数欧拉乘积的收敛区域，建立了其与素测地线定理误差项的关系，在塞尔伯格猜想下可扩展至Re s≥3/4，对平凡表示则借鉴拉马努金工作思想证明。

Generating functions for power moments of elliptic curves over $\mathbb{F}_p$

Seminal works by Birch and Ihara gave formulas for the $m$th power moments of the traces of Frobenius endomorphisms of elliptic curves over $\mathbb{F}_{p}$ for primes $p \geq 5$. Recent works by Kaplan and Petrow generalized these results to the setting of elliptic curves that contain a subgroup isomorphic to a fixed finite abelian group $A$. We revisit these formulas and determine a simple expression for the zeta function $Z_p(A; t)$, the generating function for these $m$th power moments. In particular, we find that \[ Z_p(A;t) = \frac{\widehat{Z}_p(A; t)}{\displaystyle \prod_{a \in \textrm{…

On the density function for the value-distribution of automorphic $L$-functions

The Bohr-Jessen limit theorem is a probabilistic limit theorem on the value-distribution of the Riemann zeta-function in the critical strip. Moreover their limit measure can be written as an integral involving a certaindensity function. The existence of the limit measure is now known for a quite general class of zeta-functions, but the integral expression has been proved only for some special cases (such as Dedekind zeta-functions). In this paper we give an alternative proof of the existence of the limit measure for a general setting, and then prove the integral expression, with an explicitly …

Desingularization of complex multiple zeta-functions, fundamentals of $p$-adic multiple $L$-functions, and evaluation of their special values

This paper deals with a multiple version of zeta- and L-functions both in the complex case and in the p-adic case: [I] Our motivation in the complex case is to find suitable rigorous meaning of the values of multivariable multiple zeta-functions (MZFs) at non-positive integer points. (a) We reveal that MZFs turn to be entire on the whole space after taking the desingularization. Further we show that the desingularized function is given by a suitable finite linear combination of MZFs with some arguments shifted. It is also shown that specific combinations of Bernoulli numbers attain the special…

利用模形式的超同余

Supercongruences using modular forms

该研究针对组合数生成级数中广泛存在的模素数高次幂的超同余现象，提出利用模函数与模形式可解释大部分已发现的超同余关系，为超同余研究提供了模形式视角的理论路径。

On Twisted Zeta-Functions at s=0

Let K be an abelian extension of a totally real number field k, K^+ its maximal real subfield and G=Gal(K/k). We have previously used twisted zeta-functions to define a meromorphic CG-valued function Phi_{K/k}(s) in a way similar to the use of partial zeta-functions to define the better-known function Theta_{K/k}(s). For each prime number p, we now show how the value Phi_{K/k}(0) combines with a p-adic regulator of semilocal units to define a natural Z_pG-submodule of Q_pG which we denote {frak S}_{K/k}. If p is odd and splits in k, our main theorem states that {frak S}_{K/k} is (at least) con…

Abel问题、Gauss同余与Cartier同余在数域上的推广

Abel's problem, Gauss and Cartier congruences over number fields

本文将Abel问题中基于Gauss同余的代数可解性判据推广到任意代数函数，通过引入数域上的Gauss与Cartier同余，并应用于超几何情形及Artin-Mazur zeta函数。

On derivatives of Kato's Euler system for elliptic curves

In this paper we study a new conjecture concerning Kato's Euler system of zeta elements for elliptic curves $E$ over $\mathbb{Q}$. This conjecture, which we refer to as the `Generalized Perrin-Riou Conjecture', predicts a precise congruence relation between a `Darmon-type derivative' of the zeta element of $E$ over an arbitrary real abelian field and the critical value of an appropriate higher derivative of the $L$-function of $E$ over $\mathbb{Q}$. We prove that the conjecture specializes in the relevant case of analytic rank one to recover Perrin-Riou's conjecture on the logarithm of Kato's …

An arithmetic formula for certain coefficients of the Euler product of Hecke polynomials

In 1997 the author found a criterion for the Riemann hypothesis for the Riemann zeta function, involving the nonnegativity of certain coefficients associated with the Riemann zeta function. In 1999 Bombieri and Lagarias obtained an arithmetic formula for these coefficients using the ``explicit formula'' of prime number theory. In this paper, the author obtains an arithmetic formula for corresponding coefficients associated with the Euler product of Hecke polynomials, which is essentially a product of L-functions attached to weight 2 cusp forms (both newforms and oldforms) over Hecke congruence…

Multiple Divisor Functions and Multiple Zeta Values at Levle N

Multiple zeta values (MZVs) are generalizations of Riemann zeta values at positive integers to multiple variable setting. These values can be further generalized to level $N$ multiple polylog values by evaluating multiple polylogs at $N$-th roots of unity. In this paper, we consider another level $N$ generalization by restricting the indices in the iterated sums defining MZVs to congruences classes modulo $N$, which we call the MZVs at level $N$. The goals of this paper are two-fold. First, we shall lay down the theoretical foundations of these values such as their regularizations and double s…

算术赫克三角形群轨道的密度

Densities of arithmetic Hecke triangle group orbits

研究\(q = 4\)和\(q = 6\)时算术赫克三角形群\(\Gamma_q\)轨道的渐近密度，运用初等数论技术及相关函数性质与同余条件给出新证明，发现\(q = 4\)时互质整数对集合的划分特性。

多项式对数函数的Ramaswami类型翻译公式

Ramaswami Type translation formulae for the polylogarithm functions

本文研究多项式对数函数的Ramaswami和Apostol类型翻译公式，并探讨其在伯努利数递推关系和切线数同余性质中的应用。

$k$-不可分划分中的部分在剩余类之间总是表现出偏差

Parts in $k$-indivisible Partitions Always Display Biases between Residue Classes

本文证明在 $k$-不可分划分中，随着 $n$ 增大，各部分模 $t$ 的剩余类之间总是存在偏差，不存在平局，通过 $L$-函数和非消失结果证实了先前猜想。

关于求和-幂除数函数迭代的同余关系及黎曼假设的条件推论

On Congruences for Iterates of the Sum--Power Divisor Function and Conditional Implications for the Riemann Hypothesis

本文研究了求和-幂除数函数迭代的同余关系，证明不存在整数n>1满足所有k≥1的同余条件，并探讨了其与黎曼假设的关系。

The dimensions of spaces of Siegel cusp forms of general degree

In this paper, we give a dimension formula for spaces of Siegel cusp forms of general degree with respect to neat arithmetic subgroups. The formula was conjectured before by several researchers. The dimensions are expressed by special values of Shintani zeta functions for spaces of symmetric matrices at non-positive integers. This formula was given by Shintani for only a small part of the geometric side of the trace formula. To be precise, it is the contribution of unipotent elements corresponding to the partitions $(2^j,1^{2n-2j})$, where $n$ denotes the degree and $0\leq j \leq n$. Hence, ou…

构造算术西格尔模形式：实乘法阿贝尔曲面的西塔提升与显式方法

Constructing Arithmetic Siegel Modular Forms: Theta Lifting and Explicit Methods for Real Multiplication Abelian Surfaces

该研究提出显式可计算蓝图，利用酉对偶对$(\text{U}(2,2), \text{Sp}_4)$的西塔对应构造实乘法阿贝尔曲面的向量值西格尔模形式，通过局部施瓦茨函数、分歧素数测试向量构造及计算流程，为朗兰兹函子性实例提供路线图。