On Gauss-Jacobi sums

关于Jacobi和的新恒等式及相关循环矩阵

On new identities of Jacobi sums and related cyclotomic matrices

本文利用Jacobi和的算术性质，研究涉及Jacobi和的乘积，并揭示这些乘积与特定循环矩阵之间的联系，验证了Sun在2019年提出的猜想并获得更强的结果。

On the $\ell$-adic valuation of certain Jacobi sums

Fix distinct primes $\ell$ and $f$, a finite field $\mathbf{F}_{q}$ such that $q \equiv 1 \pmod{\ell f}$, and a character $χ: \mathbf{F}_{q}^{\times} \to \mathbf{C}^{\times}$ of exact order $\ell f$. We present a new $\ell$-adic congruence for the Jacobi sum $J(χ^{\ell}, χ^{f})$. These Jacobi sums are Frobenius eigenvalues of the curve $y^{\ell} = x^{f} + 1$.

Evaluating Prime Power Gauss and Jacobi Sums

We show that for any mod $p^m$ characters, $χ_1, \dots, χ_k,$ the Jacobi sum, $$ \sum_{x_1=1}^{p^m}\dots \sum_{\substack{x_k=1\\x_1+\dots+x_k=B}}^{p^m}χ_1(x_1)\dots χ_k(x_k), $$ has a simple evaluation when $m$ is sufficiently large (for $m\geq 2$ if $p\nmid B$). As part of the proof we give a simple evaluation of the mod $p^m$ Gauss sums when $m\geq 2$.

On a generalization of Jacobi sums

We prove an estimate for multi-variable multiplicative character sums over affine subspaces of $\mathbb A^n_k$, which generalize the well known estimates for both classical Jacobi sums and one-variable polynomial multiplicative character sums.

Jacobi和的一个刻画

A characterization of Jacobi sums

本文通过三条基本性质给出有限域上 Jacobi 和的抽象刻画，证明任何满足这些性质的非平凡有限 Abel 群上的函数都自然来自某个有限域的乘法特征群及其 Jacobi 和。

Jacobi sums, Fermat Jacobians, and ranks of abelian varieties over towers of function fields

Our goal in this note is to give a number of examples of abelian varieties over function fields k(t) which have bounded ranks in towers of extensions such as k(t^{1/d}) for varying d. Along the way we prove some new results on Fermat curves which may be of independent interest.

The $p$-rank of the reduction $\rm{mod}\, p$ of jacobians and Jacobi sums

Let $Y_K\to X_K$ be a ramified cyclic covering of curves, where $K$ is a cyclotomic field. In this work we study the $p$-rank of the reduction $\rm{mod}\, p$ of a model of the jacobian of $Y_K$. In this way, we obtain counterparts of the Deuring polynomial, defined for elliptic curves, for genus greater than one. Moreover, we show that curves $Y_K$ give Hecke characters for cyclotomic fields. To carry out this study we use Jacobi sums and certain $L$-functions.

Equidistributions of Jacobi sums

Let $\mathbf{F}_q$ be a finite field of $q$ elements. We show that the normalized Jacobi sum $J(χ,η)/\sqrt{q}$, for each fixed non-trivial multiplicative character $η$, becomes equidistributed in the unit circle as $q\rightarrow+\infty,$ when $χ$ runs over all non-trivial multiplicative characters different from $η^{-1}.$ Previously, the similar equidistribution was obtained by Katz and Zheng by varying both of $χ$ and $η$. On the other hand, we also obtain the equidistribution of $J(χ,η)$ as $(χ,η)$ runs over $\mathcal{X}\times\mathcal{Y}\subseteq(\widehat{\mathrm{F}^*})^2$, as long as $|\mat…

Gauss sums, Jacobi sums and cyclotomic units related to torsion Galois modules

Let $G$ be a finite group and let $N/E$ be a tamely ramified $G$-Galois extension of number fields. We show how Stickelberger's factorization of Gauss sums can be used to determine the stable isomorphism class of various arithmetic $\mathbb{Z}[G]$-modules attached to $N/E$. If $\mathcal{O}_N$ and $\mathcal{O}_E$ denote the rings of integers of $N$ and $E$ respectively, we get in particular that $\mathcal{O}_N\otimes_{\mathcal{O}_E}\mathcal{O}_N$ defines the trivial class in the class group $\mathrm{Cl}(\mathbb{Z}[G])$ and, if $N/E$ is also assumed to be locally abelian, that the square root of…

On the distribution of multivariate Jacobi sums

Let $\mathbf{F}_q$ be a finite field of $q$ elements. We show that the normalized Jacobi sum $q^{-(m-1)/2}J(χ_1,\dots,χ_m)$ ($χ_1\dotsm χ_m$ nontrivial) is asymptotically equidistributed on the unit circle, when $χ_1\in \mathcal{A}_1,\dots, χ_m\in \mathcal{A}_m$ run through arbitrary sets of nontrivial multiplicative characters of $\mathbf{F}_q^\times$, if $\#\mathcal{A}_1\ge q^{\frac{1}{2}+ε}$, $\#\mathcal{A}_2 \ge (\log q)^{\frac{1}δ-1}$ for $ε>δ>0$ fixed and $q\to \infty$ or if $\#\mathcal{A}_1\#\mathcal{A}_2/q\to \infty$. This extends previous results of Xi, Z. Zheng, and the authors.

On the Jacobi sums over Galois rings

In this paper, we study the Jacobi sums over Galois rings of arbitrary characteristics and completely determine their absolute values, which extends the work in \cite{feng1}, where the Jacobi sums over Galois rings with characteristics a square of a prime number were discussed.

关于经典加倍法产生的雅可比和

On Jacobi sums arising from the classical doubling method

研究有限域上一般线性群或经典群相关的非阿贝尔雅可比和，对一般线性群用近藤非阿贝尔高斯和表示，对经典群及特定特征给出显式公式并证明其在几何卢茨蒂格系列上为常数，结果依赖于吉尔施 - 泽林格的乘法性结果。

A calculation of $L$-series in terms of Jacobi sums

Let us consider a cyclic extension of a function field defined over a finite field. For a character (non-trivial) of this extension, we calculate, as a linear combinations of products of Jacobi sums, the coefficients of the polynomial given by its Dirichtlet $L$-series. In the last section we show applications of this calculation.

On the distribution of Jacobi sums

Let $\mathbf{F}_q$ be a finite field of $q$ elements. For multiplicative characters $χ_1,\dots, χ_m$ of $\mathbf{F}_q^\times$, we let $J(χ_1,\dots, χ_m)$ denote the Jacobi sum. Nicholas Katz and Zhiyong Zheng showed that for $m=2$, the normalized Jacobi sum $q^{-1/2}J(χ_1,χ_2)$ ($χ_1χ_2$ nontrivial) is asymptotically equidistributed on the unit circle as $q\to \infty$, when $χ_1$ and $χ_2$ run through all nontrivial multiplicative characters of $\mathbf{F}_q^\times$. In this paper, we show a similar property for $m\ge 2$. More generally, we show that the normalized Jacobi sum $q^{-(m-1)/2}J(χ_…

Jacobi polynomials and congruences involving some higher-order Catalan numbers and binomial coefficients

In this paper, we study congruences on sums of products of binomial coefficients that can be proved by using properties of the Jacobi polynomials. We give special attention to polynomial congruences containing Catalan numbers, second-order Catalan numbers, the sequence (\seqnum{A176898}) $S_n=\frac{{6n\choose 3n}{3n\choose 2n}}{2{2n\choose n}(2n+1)},$ and the binomial coefficients ${3n\choose n}$ and ${4n\choose 2n}$. As an application, we address several conjectures of Z.\ W.\ Sun on congruences of sums involving $S_n$ and we prove a cubic residuacity criterion in terms of sums of the binomia…

涉及Jacobi和实部及相关分圆矩阵的乘积

Products involving the real parts of Jacobi sums and related cyclotomic matrices

本文研究奇素数幂q下乘积R_q(χ_q)的算术性质，该乘积与Jacobi和的实部相关，并揭示了其与分圆矩阵的联系。

Motivic Gauss 和 Jacobi 和

Motivic Gauss and Jacobi sums

本文从motives视角研究Gauss和与Jacobi和，通过定义其motivic类似物为代数对应并证明其表示Frobenius自同态，展示了Artin-Schreier曲线与Fermat簇的Chow motives同构，为经典关系提供新证明并推广了Coleman的结果。

Hyperbolic summations derived using the Jacobi functions $\text{dc}$ and $\text{nc}$

We introduce a method that is based on Fourier series expansions related to Jacobi elliptic functions and that we apply to determine new identities for evaluating hyperbolic infinite sums in terms of the complete elliptic integrals $K$ and $E$. We apply our method to determine generalizations of a family of $\text{sech}$-sums given by Ramanujan and generalizations of a family of $\text{csch}$-sums given by Zucker. Our method has the advantage of producing evaluations for hyperbolic sums with sign functions that have not previously appeared in the literature on hyperbolic sums. We apply our met…

与Jacobi椭圆函数相关的一般Berndt型积分与级数

General Berndt-Type Integrals and Series Associated with Jacobi Elliptic Functions

本文利用围道积分、双曲和及矩阵方法，研究分母具有任意正次数的一般Berndt型积分，并将其表示为含Γ^4(1/4)和π^{-1}的有理系数多项式，同时提出若干猜想。

关于p次环形数域和涉及雅可比和的环形矩阵

On $p$-th cyclotomic field and cyclotomic matrices involving Jacobi sums

本文研究了p次环形矩阵的算术性质，证明了其行列式与特定多项式系数的关系。

Davenport-Hasse's Theorem for Polynomial Gauss Sums over Finite Fields

In this paper, we study the polynomial Gauss sums over finite fields, and present an analogue of Davenport-Hasse's theorem for the entire polynomial Gauss sums, which is a generalization of the previous result obtained by Hayes.

Determinations of Analogues of Gauss Sums and Other Trigonometric Sums

Explicit determinations of several classes of trigonometric sums are given. These sums can be viewed as analogues or generalizations of Gauss sums. In a previous paper, two of the present authors considered primarily sine sums associated with primitive odd characters. In this paper, we establish two general theorems involving both sines and cosines, with more attention given to cosine sums in the several examples that we provide.

Gauss-Heilbronn 和与 Deligne-Lusztig 类曲线的覆盖

Gauss--Heilbronn Sums and Coverings of Deligne--Lusztig Type Curves

本文研究Gauss-Heilbronn和与Deligne-Lusztig型曲线的覆盖，通过分析3典型Witt向量长度为二的情况，确定相关曲线的Frobenius斜率。

Gauss sums of some matrix groups over $\Bbb Z/n\Bbb Z$

In this paper, we will explicitly calculate Gauss sums for the general linear groups and the special linear groups over $\Bbb Z_n$, where $\Bbb Z_n=\Bbb Z/n \Bbb Z$ and $n>0$ is an integer. For $r$ being a positive integer, the formulae of Gauss sums for ${\rm GL}_r(\Bbb Z_n)$ can be expressed in terms of classical Gauss sums over $\Bbb Z_n$, while the formulae of Gauss sums for ${\rm SL}_r(\Bbb Z_n)$ can be expressed in terms of hyper-Kloosterman sums over $\Bbb Z_n$. As an application, we count the number of $r\times r$ invertible matrices over $\Bbb Z_n$ with given trace by using the the fo…

素数 l 的 l^2 阶与 2l^2 阶雅可比和的计算

Computation of Jacobi sums of order l^2 and 2l^2 with prime l

本文研究奇素数 l 的 l^2 阶与 2l^2 阶雅可比和的快速计算，通过用最少数量分圆数表示雅可比和，并给出验证公式及最少性的附加算法。

Higher order moments of generalized quadratic Gauss sums weighted by $L$-functions

The main purpose of this paper is to study higher order moments of the generalized quadratic Gauss sums weighted by $L$-functions using estimates for character sums and analytic methods. We find asymptotic formulas for three character sums which arise naturally in the study of higher order moments of the generalized quadratic Gauss sums. We then use these character sum estimates to find asymptotic formulas for the $\nth{6}$ and $\nth{8}$ order moments of the generalized quadratic Gauss sums weighted by $L$-functions. Our asymptotic formulas satisfy a conjecture of Wenpeng Zhang.

The value distribution of incomplete Gauss sums

It is well known that the classical Gauss sum, normalized by the square-root number of terms, takes only finitely many values. If one restricts the range of summation to a subinterval, a much richer structure emerges. We prove a limit law for the value distribution of such incomplete Gauss sums. The limit distribution is given by the distribution of a certain family of periodic functions. Our results complement Oskolkov's pointwise bounds for incomplete Gauss sums as well as the limit theorems for quadratic Weyl sums (theta sums) due to Jurkat and van Horne and the second author.

Berndt型积分及与Ramanujan和Jacobi椭圆函数相关的级数

Berndt-Type Integrals and Series Associated with Ramanujan and Jacobi Elliptic Functions

本文以闭式求解含双曲与三角函数的Berndt型无穷积分，通过Gamma函数特殊值及椭圆函数展开求双曲和，并给出显式公式与猜想。

Short incomplete Gauss sums and rational points on metaplectic horocycles

In the present paper we investigate the limiting behaviour of short incomplete Gauss sums at random argument as the number of terms goes to infinity. We prove that the limit distribution is given by the distribution of theta sums and differs from the limit law for long Gauss sums studied by the author and Marklof. The key ingredient in the proof is an equidistribution theorem for rational points on horocycles in the metaplectic cover of SL(2,R).

Jacobi Sums and Correlations of Sidelnikov Sequences

We consider the problem of determining the cross-correlation values of the sequences in the families comprised of constant multiples of $M$-ary Sidelnikov sequences over $\mathbb{F}_q$, where $q$ is a power of an odd prime $p$. We show that the cross-correlation values of pairs of sequences from such a family can be expressed in terms of certain Jacobi sums. This insight facilitates the computation of the cross-correlation values of these sequence pairs so long as $ϕ(M)^{ϕ(M)} \leq q.$ We are also able to use our Jacobi sum expression to deduce explicit formulae for the cross-correlation distr…

四次高斯和与元裂隙θ函数

Quartic Gauss sums over primes and metaplectic theta functions

本文改进了四次高斯和的估计，并引入了新的数学工具，同时推测了相关矩的渐进行为。

A Stickleberger theorem for p-adic Gauss sums

We define "splitting functions of level l" for any integer l>0. These functions generalize Dwork's splitting functions : they allow us to represent additive characters of order $p^l$. Then we use these functions to obtain a Stickleberger theorem on the $p$-adic valuation of new Gauss sums.

On some conjectures on Generalized quadratic Gauss sums and related problems

The main purpose of this article is to study higher power mean values of generalized quadratic Gauss sums using estimates for character sums, analytic method and algebraic geometric methods. In this article, we prove two conjectures which were proposed in \cite{BLZ}.

An identity on the $2m$-th power mean value of the generalized Gauss sums

In this paper, using combinatorial and analytic methods, we prove an exact calculating formula on the $2m$-th power mean value of the generalized quadratic Gauss sums for $m\geq 2$. This solves a conjecture of He and Zhang [`On the $2k$-th power mean value of the generalized quadratic Gauss sums', Bull. Korean Math. Soc. 48 (2011), No.1, 9-15].

Gauss sums of cubic characters over $GF(p^r)$, $p$ odd

An elementary approach is shown which derives the values of the Gauss sums over $\mathbb F_{p^r}$, $p$ odd, of a cubic character without using Davenport-Hasse's theorem. New links between Gauss sums over different field extensions are shown in terms of factorizations of the Gauss sums themselves, which are then rivisited in terms of prime ideal decompositions. Interestingly, one of these results gives a representation of primes $p$ of the form $6k+1$ by a binary quadratic form in integers of a subfield of the cyclotomic field of the $p$-th roots of unity.

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。

Mock Jacobi forms in basic hypergeometric series

We show that some $q$-series such as universal mock theta functions are linear sums of theta quotients and mock Jacobi forms of weight 1/2, which become holomorphic parts of real analytic modular forms when they are restricted to torsion points and multiplied by suitable powers of $q$. And we prove that certain linear sums of $q$-series are weakly holomorphic modular forms of weight 1/2 due to annihilation of mock Jacobi forms or completion by mock Jacobi forms. As an application, we obtain a relation between the rank and crank of a partition.

Higher level quadratically twisted Gauss sums and totally isotropic subspaces

We consider a generalized Gauss sum supported on matrices over a number field. We evaluate this Gauss sum and relate it to the number of totally isotropic subspaces of related quadratic spaces. Then we consider a further generalization of such a Gauss sum, realizing its value in terms of numbers of totally isotropic subspaces of related quadratic spaces.

On equidistribution of Gauss sums of cuspidal representations of $GL_d(\mathbb F_q)$

We investigate the distribution of the angles of Gauss sums attached to the cuspidal representations of general linear groups over finite fields. In particular we show that they happen to be equidistributed w.r.t.the Haar measure. However, for representations of $PGL_2(\mathbb F_q)$, they are clustered around $1$ and $-1$ for odd $p$ and around $1$ for $p=2$.

Complete Solving for Explicit Evaluation of Gauss Sums in the Index 2 Case

Let $p$ be a prime number, $q=p^f$ for some positive integer $f$, $N$ be a positive integer such that $\gcd(N,p)=1$, and let $\k$ be a primitive multiplicative character of order $N$ over finite field $\fq$. This paper studies the problem of explicit evaluation of Gauss sums in "\textsl{index 2 case}" (i.e. $f=\f{\p(N)}{2}=[\zn:\pp]$, where $\p(\cd)$ is Euler function). Firstly, the classification of the Gauss sums in index 2 case is presented. Then, the explicit evaluation of Gauss sums $G(\k^\la) (1\laN-1)$ in index 2 case with order $N$ being general even integer (i.e. $N=2^{r}\cd N_0$ wher…

关于49阶Jacobi和$J(1, 1)_{49}$的更简单同余式

Simpler congruences for Jacobi sum $J(1, 1)_{49}$ of order 49

本文确定了有限域上49阶Jacobi和的同余式，并证明在artiad和hyperartiad素数情形下存在更简单的同余式。

Incomplete Gauss sums modulo primes

We obtain a new bound for incomplete Gauss sums modulo primes. Our argument falls under the framework of Vinogradov's method which we use to reduce the problem under consideration to bounding the number of solutions to two distinct systems of congruences. The first is related to Vinogradov's mean value theorem, although the second does not appear to have been considered before. Our bound improves on current results in the range $N\ge q^{2k^{-1/2}+O(k^{-3/2})}$.

A note on the sign (unit root) ambiguities of Gauss sums in index 2 and 4 cases

Recently, the explicit evaluation of Gauss sums in the index 2 and 4 cases have been given in several papers (see [2,3,7,8]). In the course of evaluation, the sigh (or unit root) ambiguities are unavoidably occurred. This paper presents another method, different from [7] and [8], to determine the sigh (unit root) ambiguities of Gauss sums in the index 2 case, as well as the ones with odd order in the non-cyclic index 4 case. And we note that the method in this paper are more succinct and effective than [8] and [7].

On refined metric and hermitian structures in arithmetic, I: Galois-Gauss sums and weak ramification

We use techniques of relative algebraic K-theory to develop a common refinement of the existing theories of metrized and hermitian Galois structures in arithmetic. As a first application of this very general approach, we then use it to prove several new results, and to formulate a framework of new conjectures, concerning the detailed arithmetic properties of wildly ramified Galois-Gauss sums.

Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions

In this paper we derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi's 4 and 8 squares identities to 4n^2 or 4n(n+1) squares, respectively, without using cusp forms. In fact, we similarly generalize to infinite families all of Jacobi's explicitly stated degree 2, 4, 6, 8 Lambert series expansions of classical theta functions. In addition, we extend Jacobi's special analysis of 2 squares, 2 triangles, 6 squares, 6 triangles to 12 squares, 12 triangles, 20 squares, 20 triangles, respectively. These results, depen…

Gauss Sums of the Cubic Character over $GF(2^m)$: an elementary derivation

An elementary approach is shown which derives the value of the Gauss sum of a cubic character over a finite field $\mathbb F_{2^s}$ without using Davenport-Hasse's theorem (namely, if $s$ is odd the Gauss sum is -1, and if $s$ is even its value is $-(-2)^{s/2}$).

Explicit Evaluation of Double Gauss Sums

We present an explicit evaluation of the double Gauss sum $\displaystyle G(a,b,c;S;p^n):=\sum_{x,y=0}^{p^n-1} e^{2πi S(ax^2+bxy+cy^2)/p^n}$, where $a, b, c$ are integers such that $\gcd(a,b,c)=1$, $p$ is a prime, $n$ is a positive integer, and $S$ is an integer coprime to $p$.

On the Galois-Gauss sums of weakly ramified characters

Bley, Burns and Hahn used relative algebraic $K$-theory methods to formulate a precise conjectural link between the (second Adams-operator twisted) Galois-Gauss sums of weakly ramified Artin characters and the square root of the inverse different of finite, odd degree, Galois extensions of number fields. We provide concrete new evidence for this conjecture in the setting of extensions of odd prime-power degree by using a refined version of a well-known result of Ullom.

Bilinear Forms with Kloosterman and Gauss Sums

We obtain several estimates for bilinear form with Kloosterman sums. Such results can be interpreted as a measure of cancellations amongst with parameters from short intervals. In particular, for certain ranges of parameters we improve some recent results of Blomer, Fouvry, Kowalski, Michel and Milićević and also of Fouvry, Kowalski and Michel. In particular, we improve the bound on the error term in the asymptotic formula for mixed moments of $L$-series associated with Hecke eigenforms.

Gauss Sums, Jacobi Sums, and $p$-ranks of Cyclic Difference Sets

We study quadratic residue difference sets, GMW difference sets, and difference sets arising from monomial hyperovals, all of which are $(2^d-1, 2^{d-1}-1, 2^{d-2}-1)$ cyclic difference sets in the multiplicative group of the finite field $F_{2^d}$ of $2^d$ elements, with $d \geq 2$. We show that, except for a few cases with small $d$, these difference sets are all pairwise inequivalent. This is accomplished in part by examining their 2-ranks. The 2-ranks of all of these difference sets were previously known, except for those connected with the Segre and Glynn hyperovals. We determine the 2-ra…

An explicit evaluation of $\nth{10}$-power moment of quadratic Gauss sums and some applications

In this paper we have estimated one multi-variable character sum \begin{align*} \sum_{a=2}^{p-2}\sum_{b=1}^{p-1}\sum_{c=2}^{p-2}\sum_{d=1}^{p-1}\left(\frac{a^2-b^2}{p}\right)\left(\frac{b^2-1}{p}\right) \left(\frac{c^2-d^2}{p}\right)\left(\frac{d^2-1}{p}\right)\left(\frac{a^2c^2-1}{p}\right), \end{align*} for odd prime $p$. With the help of our estimate of the above character sum, we have studied the tenth power mean value of generalized quadratic Gauss sums using estimates for character sums and analytic methods.

Improved bounds on Gauss sums in arbitrary finite fields

Let $q$ be a power of a prime and let $\mathbb{F}_q$ be the finite field consisting of $q$ elements. We establish new explicit estimates on Gauss sums of the form $S_n(a) = \sum_{x\in \mathbb{F}_q}ψ_a(x^n)$, where $ψ_a$ is a nontrivial additive character. In particular, we show that one has a nontrivial upper bound on $|S_n(a)|$ for certain values of $n$ of order up to $q^{1/2 + 1/68}$. Our results improve on the previous best known bound, due to Zhelezov.

Congruence Relations Connecting Tate-Shafarevich Groups with Bernoulli-Hurwitz Numbers by Elliptic Gauss Sums in Eisenstein Case

There are classical congruences between the class number of an imaginary quadratic field and a Bernoulli number or an Euler number. Under the BSD conjecture, Onishi obtained an elliptic generalization of these congruences, which gives congruences between the order of the Tate-Shafarevich group of certain elliptic curves with CM by the Gauss integers ring and Mordell-Weil rank 0, and a coefficient of power series expansion of an elliptic function associated to Gauss integers ring. In this paper, we provide Onishi's type congruences for the Eisenstein integers case.

多重高斯和

Multiple Gauss sums

研究多重高斯和的界，并应用于改进Birch-Goldbach问题的结果。