A Stickleberger theorem for p-adic Gauss sums

$p$-adic L-functions and Classical Congruences

In this paper, using $p$-adic analysis and $p$-adic L-functions, we show how to extend classical congruences (due to Wilson, Gauss, Dirichlet, Jacobi, Wolstenholme, Glaisher, Morley, Lemher and other people) to modulo $p^k$ for any $k>0$.

Zeros of $p$-adic hypergeometric functions, $p$-adic analogues of Kummer's and Pfaff's identities

We classify all the zeros and non-zero values of a family of hypergeometric series in the $p$-adic setting. These values of hypergeometric series in the $p$-adic setting lead to transformations of hypergeometric series in the $p$-adic setting which can be described as $p$-adic analogues of Kummer's and Pfaff's linear transformations on classical hypergeometric series. We also evaluate certain summation identities for hypergeometric series in the $p$-adic setting as well as Gaussian hypergeometric series.

Gauss-Manin connections for p-adic families of nearly overconvergent modular forms

We interpolate the Gauss-Manin connection in p-adic families of nearly overconvergent modular forms. This gives a family of Maass-Shimura type differential operators from the space of nearly overconvergent modular forms of type r to the space of nearly overconvergent modular forms of type r + 1 with p-adic weight shifted by 2. Our construction is purely geometric, using Andreatta-Iovita-Stevens and Pilloni's geometric construction of eigencurves, and should thus generalize to higher rank groups.

Sato-Tate Distribution of $p$-adic hypergeometric functions

Recently Ono, Saad and the second author \cite{KHN} initiated a study of value distribution of certain families of Gaussian hypergeometric functions over large finite fields. They investigated two families of Gaussian hypergeometric functions and showed that they satisfy semicircular and Batman distributions. Motivated by their results we aim to study distributions of certain families of hypergeometric functions in the $p$-adic setting over large finite fields. In particular, we consider two and six parameters families of hypergeometric functions in the $p$-adic setting and obtain that their l…

$\ell$-进超几何函数与结合子

The $\ell$-adic hypergeometric function and associators

本文利用 Galois 作用定义 $\ell$-进超几何函数，并通过偶酉结合子的矩阵特殊化与 gamma 函数的联系，证明其 Gauss 定理和 Euler 变换公式类似物。

p-Adic distribution of CM points and Hecke orbits. I. Convergence towards the Gauss point

We study the asymptotic distribution of CM points on the moduli space of elliptic curves over $\mathbb{C}_p$, as the discriminant of the underlying endomorphism ring varies. In contrast with the complex case, we show that there is no uniform distribution. In this paper we characterize all the sequences of discriminants for which the corresponding CM points converge towards the Gauss point of the Berkovich affine line. We also give an analogous characterization for Hecke orbits. In the companion paper we characterize all the remaining limit measures of CM points and Hecke orbits.

与$p$-进多重多对数相关的$p$-进超几何函数

$p$-adic hypergeometric function related with $p$-adic multiple polylogarithms

本文受Ohno-Zagier公式启发，构造了高斯超几何函数的$p$-进类比，并借助与$p$-进多重多对数的关系建立严格解析延拓框架，证明其$p$-进高斯超几何定理。

$p$-adic formulas and unit root $F$-subcrystals of the hypergeometric system

We define the notion of {\it Dwork family of logarithmic $F$-crystals}, a typical example of which is the family of Gauss hypergeometricdifferential systems, viewed as parametrized by their exponents of algebraic monodromy. The $p$-adic analytic dependence of the Frobenius operation upon those exponents, is Dwork's "Boyarsky Principle". We discuss, in favorable cases, the $p$-adic analytic continuation of the unit root $F$-subcrystal in the open tube of a singularity, uniformly w.r.t. the exponents. We obtain a conceptual proof of the Koblitz-Diamond formula $p$-adically analog to Gauss' evalu…

Values of $p$-adic hypergeometric functions, and $p$-adic analogue of Kummer's linear identity

Let $p$ be an odd prime and $\mathbb{F}_p$ be the finite field with $p$ elements. This paper focuses on the study of values of a generic family of hypergeometric functions in the $p$-adic setting which we denote by ${_{3n-1}G_{3n-1}}(p, t),$ where $n\geq1$ and $t\in\mathbb{F}_p$. These values are expressed in terms of numbers of zeros of certain polynomials over $\mathbb{F}_p$. These results lead to certain $p$-adic analogues of classical hypergeometric identities. Namely, we obtain $p$-adic analogues of particular cases of a Gauss' theorem and a Kummer's theorem. Moreover, we examine the zero…

函数域上的几何高斯和与Gross-Koblitz公式

Geometric Gauss Sums and Gross-Koblitz Formula over Function Fields

本文在正特征函数域上引入几何高斯和，通过证明Gross-Koblitz型公式将其与$v$-进伽马值联系，从而导出反射公式、Stickelberger定理等基本性质。

The 4-Adic Complexity of Interleaved Quaternary Sequences of Even Length with Optimal Autocorrelation

Su et al. proposed several new classes of quaternary sequences of even length with optimal autocorrelation interleaved by twin-prime sequences pairs, GMW sequences pairs or binary cyclotomic sequences of order four in \cite{S1}. In this paper, we determine the 4-adic complexity of these quaternary sequences with period $2n$ by using correlation function and the "Gauss periods" of order four and "quadratic Gauss sums" on finite field $\mathbb{F}_n$ and valued in $\mathbb{Z}^{*}_{4^{2n}-1}$. Our results show that they are safe enough to resist the attack of the rational approximation algorithm.

A unipotent circle action on $p$-adic modular forms

Following a suggestion of Peter Scholze, we construct an action of $\hat{\mathbb{G}}_m$ on the Katz moduli problem, a profinite-étale cover of the ordinary locus of the $p$-adic modular curve whose ring of functions is Serre's space of $p$-adic modular functions. This action is a local, $p$-adic analog of a global, archimedean action of the circle group $S^1$ on the lattice-unstable locus of the modular curve over $\mathbb{C}$. To construct the $\hat{\mathbb{G}}_m$-action, we descend a moduli-theoretic action of a larger group on the (big) ordinary Igusa variety of Caraiani-Scholze. We compute…

The Gauss higher relative class number problem

Assuming the 2-adic Iwasawa main conjecture, we find all CM fields with higher relative class number at most 16: there are at least 31 and at most 34 such fields, and exactly one is not abelian.

Zeros of hypergeometric functions in the $p$-adic setting

Let $p$ be an odd prime and $\mathbb{F}_p$ be the finite field with $p$ elements. McCarthy \cite{mccarthy-pacific} initiated a study of hypergeometric functions in the $p$-adic setting. This function can be understood as $p$-adic analogue of Gauss' hypergeometric function, and also some kind of extension of Greene's hypergeometric function over $\mathbb{F}_p$. In this paper we investigate values of two generic families of McCarthy's hypergeometric functions denoted by ${_nG_n}(t)$, and ${_n\widetilde{G}_n}(t)$ for $n\geq3$, and $t\in\mathbb{F}_p$. The values of the function ${_nG_n}(t)$ certai…

Gaussian distribution of short sums of trace functions over finite fields

We show that under certain general conditions, short sums of $\ell$-adic trace functions over finite fields follow a normal distribution asymptotically when the origin varies, generalizing results of Erdős-Davenport, Mak-Zaharescu and Lamzouri. In particular, this applies to exponential sums arising from Fourier transforms such as Kloosterman sums or Birch sums, as we can deduce from the works of Katz. By approximating the moments of traces of random matrices in monodromy groups, a quantitative version can be given as in Lamzouri's article, exhibiting a different phenomenon than the averaging …

On inverted Kloosterman sums over finite fields

The classical $n$-variable Kloosterman sums over finite fields are well understood by Deligne's theorem from complex point of view and by Sperber's theorem from $p$-adic point of view. In this paper, we study the complex and $p$-adic estimates of inverted $n$-variable Kloosterman sums, addressing a question of N. Katz (1995). We shall give two complex estimates. The first one is elementary based on Gauss sums. The second estimate is deeper, depending on the cohomological results of Adolphson-Sperber, Denef-Loeser and Fu for twisted toric exponential sums. This deeper result assumes that the ch…

Hilbert模形式的有限斜率族的扭曲三重积$p$-adic $L$-函数

Twisted Triple Product $p$-adic $L$-function for Finite Slope Families of Hilbert Modular Forms

本文构造了Hilbert模簇上相对de Rham上同调类的过收敛族，证明其Gauss-Manin联络的$p$-adic迭代收敛性，并据此构造了Hilbert模形式有限斜率族的$p$-adic扭曲三重积$L$-函数。

Gauss--Manin 联络延拓到全纯余容许 D-cap-模

Gauss--Manin connections extend to holonomic coadmissible D-cap-modules

本文证明光滑刚性解析簇上 Gauss--Manin 联络沿 Zariski 开浸入的推出是余容许且全纯的，推广了经典正则奇点定理，并借助 p-adic 方法与对数联络延拓性建立结论。

Metaplectic Covers of $p$-adic Groups and Quantum Groups at Roots of Unity

We describe the structure of the Whittaker or Gelfand-Graev module on a $n$-fold metaplectic cover of a $p$-adic group $G$ at both the Iwahori and spherical level. We express our answer in terms of the representation theory of a quantum group at a root of unity attached to the Langlands dual group of $G$. To do so, we introduce an algebro-combinatorial model for these modules and develop for them a Kazhdan-Lusztig theory involving new generic parameters. These parameters can either be specialized to Gauss sums to recover the $p$-adic theory or to the natural grading parameter in the representa…

$p$-进四元数 Hermite 型的局部密度

Local densities of $p$-adic quaternion hermitian forms

本文研究 $p$-进四元数 Hermite 型的局部密度，证明缩放双曲平面时密度函数的线性无关性，并构造显式多项式使相关形式幂级数化为多项式。

区分高斯和

Distinguishing Gauss sums

本文研究有限域上高斯和相等何时蕴含特征为 Frobenius 共轭，利用 Stickelberger 分解给出可检验判据，并证明阶为 q-1 或 (q-1)/2 时成立。

Converse Theorem Meets Gauss Sums (with an appendix by Zhiwei Yun)

This paper verifies $n\times 1$ Local Converse Theorem for twisted gamma factors of irreducible cuspidal representations of ${\rm GL}_n({\mathbb F}_p)$, for $n\leq 5,$ and of irreducible generic representations, for $n<\frac{q-1}{2\sqrt{q}}+1$ in the appendix by Zhiwei Yun, where $p$ is a prime and q is a power of $p$. The counterpart of $n\times 1$ converse theorem for level zero cuspidal representations also follows the established relation between gamma factors of ${\rm GL}_n({\mathcal F})$ and that of ${\rm GL}_n({\mathbb F}_q)$, where ${\mathcal F}$ denotes a $p$-adic field whose residue …

Roots of $L$-functions of characters over function fields, generic linear independence and biases

We first show joint uniform distribution of values of Kloosterman sums or Birch sums among all extensions of a finite field $\mathbb{F}_q$, for almost all couples of arguments in $\mathbb{F}_q^\times$, as well as lower bounds on differences. Using similar ideas, we then study the biases in the distribution of generalized angles of Gaussian primes over function fields and primes in short intervals over function fields, following recent works of Rudnick--Waxman and Keating--Rudnick, building on cohomological interpretations and determinations of monodromy groups by Katz. Our results are based on…

The Restricted Partition and q-Partial Fractions

The restricted partition function $p_{N}(n)$ counts the partitions of $n$ into at most $N$ parts. In the nineteenth century Sylvester showed that these partitions can be expressed as a sum of $k$-periodic quasi-polynomials ($1\leq k\leq N$) which he termed as Waves. It is now well-known that one can easily perform a wave decomposition using a special type of partial fraction decomposition (the so-called $q$-partial fractions) of the generating function of $p_{N}(n)$. In this paper we show that the coefficients of these $q$-partial fractions can be expressed as a linear combination of the Raman…

Independent linear forms on the group $Ω_p$

Let $Ω_p$ be the group of $p$-adic numbers, $ ξ_1$, $ξ_2$, $ξ_3$ be independent random variables with values in $Ω_p$ and distributions $μ_1$, $μ_2$, $μ_3$. Let $α_j, β_j, γ_j$ be topological automorphisms of $Ω_p$. We consider linear forms $L_1 = α_1ξ_1 + α_2 ξ_2+α_3 ξ_3$, $L_2=β_1ξ_1 + β_2 ξ_2+ β_3 ξ_3$ and $L_3=γ_1ξ_1 + γ_2 ξ_2+ γ_3 ξ_3$. Assuming that the linear forms $L_1$, $L_2$ and $L_3$ are independent, we describe possible distributions $μ_1$, $μ_2$, $μ_3$. This theorem is an analogue of the well-known Skitovich-Darmois theorem, where a Gaussian distribution on the real line is charac…

具有单项式形变的对角超曲面上 $\mathbb{F}_q$-点的数量

The number of $\mathbb{F}_q$-points on diagonal hypersurfaces with monomial deformation

本文研究了具有单项式形变的对角超曲面上的 $\mathbb{F}_q$-点数量，利用 Gauss 和、Jacobi 和及 $p$-进超几何函数给出了计数公式，推广了 Koblitz、Sulakashna 与 Barman 以及作者本人的已有结果。