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The modern study of the exponential sums is mainly about their analytic estimates as complex numbers, which is local. In this paper, we study one global property of the exponential sums by viewing them as algebraic integers. For a kind of generalized Kloosterman sums, we present their degrees as algebraic integers. [ABSTRACT FROM AUTHOR]

Kim, Chung, No and Chung introduced new families of M-ary sequences by using Sidel'nikov sequences and the shift-and-add method. Chung, No and Chung constructed more families of M-ary sequences based on Sidel'nikov sequences and the shift-and-inverse method. In this paper we further study the balancedness of these sequences and show that they have asymptotical uniform pattern distributions under some assumptions on the parameters of the sequences. Linear complexity profiles of the sequences are also studied. [ABSTRACT FROM AUTHOR]

A Diophantine m-tuple over a finite field F q is a set { a 1 , ... , a m } of m distinct elements in F q ∗ such that a i a j + 1 is a square in F q whenever i ≠ j . In this paper, we study M(q), the maximum size of a Diophantine tuple over F q , assuming the characteristic of F q is fixed and q → ∞ . By explicit constructions, we improve the lower bound on M(q). In particular, this improves a recent result of Dujella and Kazalicki by a multiplicative factor. [ABSTRACT FROM AUTHOR]

We prove sum representations of Appell-Lauricella functions over a finite field using confluent hypergeometric functions over the finite field. As an application, we also prove transformation formulas, summation formulas and reduction formulas for Appell-Lauricella functions over the finite field. [ABSTRACT FROM AUTHOR]

We examine hypergeometric functions in the finite field, p-adic and classical settings. In each setting, we prove a formula which splits the hypergeometric function into a sum of lower order functions whose arguments differ by roots of unity. We provide multiple applications of these results, including new reduction and summation formulas for finite field hypergeometric functions, along with classical analogues; evaluations of special values of these functions which apply in both the finite field and p-adic settings; and new relations to Fourier coefficients of modular forms. [ABSTRACT FROM AUTHOR]

Let n = 2 s p 1 α 1 ⋯ p k α k , where s = 0 or 1, α i ≥ 1 , and the distinct primes p i satisfy p i ≡ 1 (mod 4) for all i = 1 , … , k . Let Z n ∗ denote the group of units in the commutative ring Z n . In a recent paper, we defined the Paley-type graph G n of order n as the graph whose vertex set is Z n and xy is an edge if x - y ≡ a 2 (mod n) for some a ∈ Z n ∗ . Computing the number of cliques of a particular order in a Paley graph or its generalizations has been of considerable interest. In our recent paper, for primes p ≡ 1 (mod 4) and α ≥ 1 , by evaluating certain character sums, we found the number of cliques of order 3 in G p α and expressed the number of cliques of order 4 in G p α in terms of Jacobi sums. In this article we give combinatorial proofs and find the number of cliques of orders 3 and 4 in G n for all n for which the graph is defined. [ABSTRACT FROM AUTHOR]

Let k ≥ 2 be an even integer. Let q be a prime power such that q ≡ k + 1 (mod 2 k) . We define the k-th power Paley digraph of order q, G k (q) , as the graph with vertex set F q where a → b is an edge if and only if b - a is a k-th power residue. This generalizes the ( k = 2 ) Paley Tournament. We provide a formula, in terms of finite field hypergeometric functions, for the number of transitive subtournaments of order four contained in G k (q) , K 4 (G k (q)) , which holds for all k. We also provide a formula, in terms of Jacobi sums, for the number of transitive subtournaments of order three contained in G k (q) , K 3 (G k (q)) . In both cases, we give explicit determinations of these formulae for small k. We show that zero values of K 4 (G k (q)) (resp. K 3 (G k (q)) ) yield lower bounds for the multicolor directed Ramsey numbers R k 2 (4) = R (4 , 4 , … , 4) (resp. R k 2 (3) ). We state explicitly these lower bounds for k ≤ 10 and compare to known bounds, showing improvement for R 2 (4) and R 3 (3) . Combining with known multiplicative relations we give improved lower bounds for R t (4) , for all t ≥ 2 , and for R t (3) , for all t ≥ 3 . [ABSTRACT FROM AUTHOR]

For a , b , c , d , e ∈ Q × , denote by C a , b , c , d , e the nonsingular projective curve over Q given by the affine equation C a , b , c , d , e : a x 2 + b y 2 = c + d x y + e x 2 y 2. In this paper, we find a relation between the number of points on C a , b , c , d , e over F q and the finite field Appell series F 4 ∗ . As a consequence, in view of certain transformations for finite field Appell series F 4 ∗ , we prove the existence of isogenies between C a , b , c , d , ab c and the twisted Edward family of elliptic curves E α , 16 a b α d 2 . [ABSTRACT FROM AUTHOR]

Inspired by two entries published in Ramanujan's lost notebook on p. 355, Berndt, Kim, and Zaharescu presented Riesz sum identities for the twisted divisor sums. Subsequently, Kim derived analogous results by replacing twisted divisor sums with twisted sums of divisor functions. In a recent work, the authors of the present paper deduced Cohen-type identities and Voronoï summation formulas associated with these twisted sums of divisor functions. The present paper aims to derive an equivalent version of the results offered in the previous article in terms of identities involving finite sums of trigonometric functions and the doubly infinite series. As an application, the authors provide an identity for r 6 (n) , which is analogous to Hardy's famous result where r 6 (n) denotes the number of representations of natural number n as a sum of six squares. [ABSTRACT FROM AUTHOR]

Let F q be a finite field with q elements, where q is a power of a prime p. In this paper, we obtain an improvement on Weil bounds for character sums associated to a polynomial f(x) over F q , which extends the results of Wan et al. (Des. Codes Cryptogr. 81, 459–468, 2016) and Wu et al. (Des. Codes Cryptogr. 90, 2813–2821, 2022). [ABSTRACT FROM AUTHOR]

For a prime p ≡ 3 (mod 4) and a positive integer t, let q = p 2 t . Let g be a primitive element of the finite field F q . The Peisert graph P ∗ (q) is defined as the graph with vertex set F q where ab is an edge if and only if a - b ∈ ⟨ g 4 ⟩ ∪ g ⟨ g 4 ⟩ . We provide a formula, in terms of finite field hypergeometric functions, for the number of complete subgraphs of order four contained in P ∗ (q) . We also give a new proof for the number of complete subgraphs of order three contained in P ∗ (q) by evaluating certain character sums. The computations for the number of complete subgraphs of order four are quite tedious, so we further give an asymptotic result for the number of complete subgraphs of any order m in Peisert graphs. [ABSTRACT FROM AUTHOR]

We prove an exact formula for the variance of the divisor function over short intervals in A : = F q [ T ] , where q is a prime power; and for correlations of the form d (A) d (A + B) , where we average both A and B over certain intervals in A . We also obtain an exact formula for correlations of the form d (K Q + N) d (N) , where Q is prime and K and N are averaged over certain intervals with deg N ≤ deg Q - 1 ≤ deg K ; and we demonstrate that d (K Q + N) and d(N) are uncorrelated. We generalize our results to σ z defined by σ z (A) : = ∑ E ∣ A | A | z for all monics A ∈ A . Our approach is to use the orthogonality relations of additive characters on F q to translate the problems to ones involving the ranks of Hankel matrices over F q . We prove several results regarding the rank and kernel structure of these matrices, thus demonstrating their number-theoretic properties. We also discuss extending our method to other divisor sums, such as those involving d k . [ABSTRACT FROM AUTHOR]

Let F q n be a finite field with qn elements. For a positive divisor r of q n − 1 , the element α ∈ F q n * is called r-primitive if its multiplicative order is (q n − 1) / r . Also, for a non-negative integer k, the element α ∈ F q n is k-normal over F q if gcd (α x n − 1 + α q x n − 2 + ⋯ + α q n − 2 x + α q n − 1 , x n − 1) in F q n [ x ] has degree k. In this paper we discuss the existence of elements in arithmetic progressions { α , α + β , α + 2 β , ... α + (m − 1) β } ⊂ F q n with α + (i − 1) β being ri-primitive and at least one of the elements in the arithmetic progression being k-normal over F q . We obtain asymptotic results for general k , r 1 , ... , r m and concrete results when k = r i = 2 for i ∈ { 1 , ... , m } . [ABSTRACT FROM AUTHOR]

We give a definition of generalized hypergeometric functions over finite fields using modified Gauss sums, which enables us to find clear analogy with classical hypergeometric functions over the complex numbers. We study their fundamental properties and prove summation formulas, transformation formulas and product formulas. An application to zeta functions of K3-surfaces is given. In the appendix, we give an elementary proof of the Davenport–Hasse multiplication formula for Gauss sums. [ABSTRACT FROM AUTHOR]

We study covering sumsets of a prime field based on its multiplicative structure. By developing various sufficient analytic and algebraic criteria for their existence, it is shown that covering sumsets arise in two main families, namely in the form of complementary sumsets and in the form of double sumsets. In each case, the abundance of covering sumsets is supported by providing asymptotically growing lower bounds on their number which in turn point out a rich array of fruitful connections to seemingly unrelated topics such as the Titchmarsh divisor problem, Mersenne primes, Fermat quotients, partitions into cycles, quadratic reciprocity, Gauss and Jacobi sums, and density results in class field theory resulting from Chebotarev's theorem. Moreover, representations of an element taken from a prime field, in terms of the sums in a covering sumset, furnish us with new formulas for the class numbers of quadratic fields, Bernoulli numbers and Bernoulli polynomials. In this way, curious tendencies among the number of representations are discovered over half intervals. Lastly, our findings show in different circumstances that the summands of a covering sumset can seldom form an arithmetic progression, thereby indicating a tension between additive and multiplicative structures in a prime field. [ABSTRACT FROM AUTHOR]

Let p ≡ 1 (mod 3) be a prime. We study several topics on additive decompositions concerning the set C p of all non-zero cubes in the finite field of p elements. For example, we show that when p > 184291 , the set C p has no decomposition of the form C p = A + B + C with | A | , | B | , | C | ≥ 2 . [ABSTRACT FROM AUTHOR]

McCarthy (Pac J Math 261(1):219–236, 2013) defined hypergeometric functions in the p-adic setting over finite fields using p-adic gamma functions. These functions possess many properties that are analogous to classical hypergeometric type identities. In this paper, we investigate values of two generic families of these hypergeometric functions that we denote by n G n (t) p and n G ~ n (t) p for n ≥ 3 , and t ∈ F p , the finite field with p elements. These results generalize special cases of p-adic analogues of Whipple's theorem and Dixon's theorem of classical hypergeometric series. We also examine zeros of the functions n G n (t) p , and n G ~ n (t) p over F p . Moreover, we classify the values of t for which n G n (t) p = 0 for infinitely many primes. For example, we show that there are infinitely many primes for which 2 k G 2 k (- 1) p = 0 . In contrast, for t ≠ 0 there are no primes for which 2 k G ~ 2 k (t) p = 0 . [ABSTRACT FROM AUTHOR]

In this paper we characterize the c-Boomerang Connectivity Table (BCT), c ≠ 0 (thus, including the classical c = 1 case), for all monomial function x d in terms of characters and Weil sums on the finite field F p n , for an odd prime p. We further simplify these expressions for the Gold functions x p k + 1 for all 1 ≤ k < n , and p odd. It is the first such attempt for a complete description for the classical BCT and its relative c-BCT, for all parameters involved, albeit in terms of characters. [ABSTRACT FROM AUTHOR]

In the 1960s, Birch proved that the traces of Frobenius for elliptic curves taken at random over a large finite field is modeled by the semicircular distribution (i.e. the usual Sato–Tate for non-CM elliptic curves). In analogy with Birch's result, a recent paper by Ono, the author, and Saikia proved that the limiting distribution of the normalized Frobenius traces A λ ⁢ (p) of a certain family of K ⁢ 3 surfaces X λ with generic Picard rank 19 is the O ⁢ (3) distribution. This distribution, which we denote by 1 4 ⁢ π ⁢ f ⁢ (t) , is quite different from the semicircular distribution. It is supported on [ - 3 , 3 ] and has vertical asymptotes at t = ± 1 . Here we make this result explicit. We prove that if p ≥ 5 is prime and - 3 ≤ a < b ≤ 3 , then | # ⁢ { λ ∈ 픽 p : A λ ⁢ (p) ∈ [ a , b ] } p - 1 4 ⁢ π ⁢ ∫ a b f ⁢ (t) ⁢ 푑 t | ≤ 98.28 p 1 / 4 . As a consequence, we are able to determine when a finite field 픽 p is large enough for the discrete histograms to reach any given height near t = ± 1 . To obtain these results, we make use of the theory of Rankin–Cohen brackets in the theory of harmonic Maass forms. [ABSTRACT FROM AUTHOR]

Kampé de Fériet hypergeometric functions are two-variable hypergeometric functions, which are a generalization of Appell's functions. It is known that they satisfy many reduction and summation formulas. In this paper, we define Kampé de Fériet hypergeometric functions over finite fields and show analogous formulas. [ABSTRACT FROM AUTHOR]

In this paper, we present the fast computational algorithms for the Jacobi sums of orders l 2 and 2 l 2 with odd prime l by formulating them in terms of the minimum number of cyclotomic numbers of the corresponding orders. We also implement two additional algorithms to validate these formulae, which are also useful for the demonstration of the minimality of cyclotomic numbers required. [ABSTRACT FROM AUTHOR]

We describe an algorithm for computing, for all primes $p \leq X$, themod-$p$ reduction of the trace of Frobenius at $p$ of a fixed hypergeometricmotive in time quasilinear in $X$. This combines the Beukers--Cohen--Mellittrace formula with average polynomial time techniques of Harvey et al.

In this paper we obtain further improvement of index bounds for character sums of polynomials over finite fields. We present some examples, which show that our new bound is an improved bound compared to both the Weil bound and the index bound given by Wan and Wang. As an application, we give an estimation of the number of all the solutions of some algebraic curves by using our result. [ABSTRACT FROM AUTHOR]

Recently Ono, Saad and the second author [21] 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 limiting distributions are semicircular over large finite fields. In the process of doing this we also express the traces of pth Hecke operators acting on the spaces of cusp forms of even weight k ≥ 4 and levels 4 and 8 in terms of p-adic hypergeometric function which is of independent interest. These results can be viewed as p-adic analogous of some trace formulas of [1-2, 6]. [ABSTRACT FROM AUTHOR]

The hull of a linear code over finite fields is the intersection of the code and its dual, which was introduced by Assmus and Key. In this paper, we develop a method to construct linear codes with trivial hull (LCD codes) and one-dimensional hull by employing the positive characteristic analogues of Gauss sums. These codes are quasi-abelian, and sometimes doubly circulant. Some sufficient conditions for a linear code to be an LCD code (resp. a linear code with one-dimensional hull) are presented. It is worth mentioning that we present a lower bound on the minimum distances of the constructed linear codes. As an application, using these conditions, we obtain some optimal or almost optimal LCD codes (resp. linear codes with one-dimensional hull) with respect to the online Database of Grassl. [ABSTRACT FROM AUTHOR]

Here, we give a complete description of the entire c-Boomerang Connectivity Table for the Gold function over finite fields of even characteristic, by using double Weil sums. As a by-product, we generalize a result of Boura and Canteaut (IACR Trans. Symmetric Cryptol. 2018(3) : 290–310, 2018) for the classical boomerang uniformity (see also the extended abstract by Eddahmani and Mesnager at the Boolean Functions and their Applications (BFA 2021) conference). [ABSTRACT FROM AUTHOR]

We prove two transformations for the p-adic hypergeometric function, which can be described as analogues of a transformation of Euler and a transformation of Clausen. We first evaluate certain character sums and then relate them to the p-adic hypergeometric functions to deduce the transformations. We use a character sum identity proved by Ahlgren, Ono, and Penniston to deduce the p-adic Clausen's transformation. We also deduce special values of certain p-adic hypergeometric functions. [ABSTRACT FROM AUTHOR]

Many product formulas are known classically for generalized hypergeometric functions over the complex numbers. In this paper, we establish some analogous formulas for generalized hypergeometric functions over finite fields. [ABSTRACT FROM AUTHOR]

In this article, we obtain finite field analogues of classical summation identities connecting F 3 -Appell series and 4 F 3 -classical hypergeometric series. As an application, we establish a new summation formula satisfied by the 4 F 3 -Gaussian hypergeometric series. We further express certain special values of 4 F 3 -Gaussian hypergeometric series in terms of traces of the Frobenius endomorphisms of certain families of elliptic curves. We also explicitly find some special values of 4 F 3 -Gaussian hypergeometric series. [ABSTRACT FROM AUTHOR]

This paper explores the existence and distribution of primitive elements in finite field extensions with prescribed traces in several intermediate field extensions. Our main result provides an inequality-like condition to ensure the existence of such elements. We then derive concrete existence results for a special class of intermediate extensions. [ABSTRACT FROM AUTHOR]

In this paper, we develop a finite field analogue for one of the Lauricella series, F A (n) . Extending results of Greene, a finite field analogue for the multinomial coefficient is developed in order to express the Lauricella series in terms of binomial coefficients. We have further deduced certain transformation and reduction formulas for the Lauricella series F A (n) . Finally, we have obtained a number of generating functions for the Lauricella series F A (n) . [ABSTRACT FROM AUTHOR]

In this article we find finite field analogues of certain transformations satisfied by the classical hypergeometric series. Using properties of Gauss and Jacobi sums we evaluate certain character sums to establish these transformations. We then use these transformations to evaluate explicitly several special values of 2 F 1 hypergeometric functions over finite fields. Certain special values of 2 F 1 hypergeometric functions over finite fields containing trivial and quadratic characters obtained by Ono follow from our special values of 2 F 1 hypergeometric functions containing arbitrary characters as parameters. One of the finite field analogues of algebraic hypergeometric identities given by Fuselier, Long, Ramakrishna, Swisher, and Tu also follows from one of our transformation formulas. [ABSTRACT FROM AUTHOR]

We study the elliptic curves given by y 2 = x 3 + b x + t 3 n + 1 over global function fields of characteristic 3 ; in particular we perform an explicit computation of the L-function by relating it to the zeta function of a certain superelliptic curve u 3 + b u = v 3 n + 1 . In this way, using the Néron–Tate height on the Mordell–Weil group, we obtain lattices in dimension 2 · 3 n for every n ≥ 1 , which improve on the currently best known sphere packing densities in dimensions 162 (case n = 4 ) and 486 (case n = 5 ). For n = 3 , the construction has the same packing density as the best currently known sphere packing in dimension 54, and for n = 1 it has the same density as the lattice E 6 in dimension 6. [ABSTRACT FROM AUTHOR]

For non-negative integers l 1 , l 2 , ... , l n , we define character sums φ (l 1 , l 2 , ... , l n) and ψ (l 1 , l 2 , ... , l n) over a finite field which are generalizations of Jacobsthal and modified Jacobsthal sums, respectively. We express these character sums in terms of Greene's finite field hypergeometric series. We then express the number of points on the hyperelliptic curves y 2 = (x m + a) (x m + b) (x m + c) and y 2 = x (x m + a) (x m + b) (x m + c) over a finite field in terms of the character sums φ (l 1 , l 2 , l 3) and ψ (l 1 , l 2 , l 3) , and finally obtain expressions in terms of the finite field hypergeometric series. [ABSTRACT FROM AUTHOR]

In this article, we find finite field analogues of certain identities satisfied by the classical 4 F 3 -hypergeometric series and Appell series. As an application, we find new summation and product formulas satisfied by the Gaussian hypergeometric series. For example, we express a 4 F 3 -Gaussian hypergeometric series as a sum of two 2 F 1 -Gaussian hypergeometric series. We also find two identities expressing 4 F 3 -Gaussian hypergeometric series as a product of two 2 F 1 -Gaussian hypergeometric series. As another application, we find a special value of a 4 F 3 -Gaussian hypergeometric series using the summation formula. [ABSTRACT FROM AUTHOR]

Let k ≥ 2 be an integer. Let q be a prime power such that q ≡ 1 (mod k) if q is even, or, q ≡ 1 (mod 2 k) if q is odd. The generalized Paley graph of order q, G k (q) , is the graph with vertex set F q where ab is an edge if and only if a - b is a kth power residue. We provide a formula, in terms of finite field hypergeometric functions, for the number of complete subgraphs of order four contained in G k (q) , K 4 (G k (q)) , which holds for all k. This generalizes the results of Evans, Pulham and Sheehan on the original ( k = 2 ) Paley graph. We also provide a formula, in terms of Jacobi sums, for the number of complete subgraphs of order three contained in G k (q) , K 3 (G k (q)) . In both cases, we give explicit determinations of these formulae for small k. We show that zero values of K 4 (G k (q)) (resp. K 3 (G k (q)) ) yield lower bounds for the multicolor diagonal Ramsey numbers R k (4) = R (4 , 4 , ... , 4) (resp. R k (3) ). We state explicitly these lower bounds for small k and compare to known bounds. We also examine the relationship between both K 4 (G k (q)) and K 3 (G k (q)) , when q is prime, and Fourier coefficients of modular forms. [ABSTRACT FROM AUTHOR]