hyper-Kloosterman Sums Research Articles

Fourth power mean of the general s-dimensional Kloosterman sum mod p

In this article, we prove an asymptotic formula for the fourth power mean of a general s-dimensional hyper-Kloosterman sum. We find the number of solutions of certain congruence equations mod p which play an integral part to prove our main result. We use estimates for character sums and analytic methods to prove our theorem.

Hyperkloosterman sums revisited

We return to some past studies of hyperkloosterman sums ([9,10]) via p-adic cohomology with an aim to improve earlier results. In particular, we work here with Dwork's θ∞-splitting function and a better choice of basis for cohomology. To a large extent, we are guided to this choice of basis by our recent work on the p-integrality of coefficients of A-hypergeometric series [3]. In the earlier work, congruence estimates were limited to p>n+2. We are here able to remove all characteristic restrictions from earlier results.

Stratification and averaging for exponential sums: bilinear forms with generalized Kloosterman sums

We prove non-trivial bounds for bilinear forms with hyper-Kloosterman sums with characters modulo a prime $q$ which, for both variables of length $M$, are non-trivial as soon as $M\geq q^{3/8+\delta}$ for any $\delta>0$. This range, which matches Burgess's range, is identical with the best results previously known only for simpler exponentials of monomials. The proof combines refinements of the analytic tools from our previous paper and new geometric methods. The key geometric idea is a comparison statement that shows that even when the "sum-product" sheaves that appear in the analysis fail to be irreducible, their decomposition reflects that of the "input" sheaves, except for parameters in a high-codimension subset. This property is proved by a subtle interplay between \'etale cohomology in its algebraic and diophantine incarnations. We prove a first application concerning the first moment of a family of $L$-functions of degree $3$.

Exponential Sums Over Finite Fields and the Large Sieve

Abstract By using a variant of the large sieve for Frobenius in compatible systems developed in [24] and [27], we obtain zero-density estimates for arguments of $\ell $-adic trace functions over finite fields with values in some algebraic subsets of the cyclotomic integers, when the monodromy groups are known. This applies in particular to hyper-Kloosterman sums and general exponential sums considered by Katz.

Distribution questions for trace functions with values in cyclotomic integers and their reductions

We consider ℓ \ell -adic trace functions over finite fields taking values in cyclotomic integers, such as characters and exponential sums. Through ideas of Deligne and Katz, we explore probabilistic properties of the reductions modulo a prime ideal, exploiting especially the determination of their integral monodromy groups. In particular, this gives a generalization of a result of Lamzouri-Zaharescu on the distribution of short sums of the Legendre symbol reduced modulo an integer to all multiplicative characters and to hyper-Kloosterman sums.

INTEGRAL MONODROMY GROUPS OF KLOOSTERMAN SHEAVES

We show that integral monodromy groups of Kloosterman -adic sheaves of rank on are as large as possible when the characteristic is large enough, depending only on the rank. This variant of Katz's results over was known by works of Gabber, Larsen, Nori and Hall under restrictions such as large enough depending on with an ineffective constant, which is unsuitable for applications. We use the theory of finite groups of Lie type to extend Katz's ideas, in particular the classification of maximal subgroups of Aschbacher and Kleidman–Liebeck. These results will apply to study hyper-Kloosterman sums and their reductions in forthcoming work.

The Balanced Voronoi Formulas for $\textrm{GL}(n)$

Abstract In this article, we show how the $\textrm{GL}(N)$ Voronoi summation formula of [13] can be rewritten to incorporate hyper-Kloosterman sums of various dimensions on both sides. This generalizes a formula for $\textrm{GL}(4)$ with ordinary Kloosterman sums on both sides that was used in [1] to prove nonvanishing of GL(4) $L$-functions by GL(2)-twists, and later by the second-named author in [16].

Bilinear forms with Kloosterman sums and applications

We prove non-trivial bounds for general bilinear forms in hyper-Kloosterman sums when the sizes of both variables may be below the P\'olya-Vinogradov range. We then derive applications to the second moment of holomorphic cusp forms twisted by characters modulo primes, and to the distribution in arithmetic progressions to large moduli of certain Eisenstein-Hecke coefficients on $\GL_3$. Our main tools are new bounds for certain complete sums in three variables over finite fields, proved using methods from algebraic geometry, especially $\ell$-adic cohomology and the Riemann Hypothesis.

Hyper-Kloosterman Sums of Different Moduli and Their Applications to Automorphic Forms for $\operatorname{SL}_m(\mathbb{Z})$

Hyper-Kloosterman sums of different moduli appear naturally in Voronoi's summation formula for cusp forms for $\operatorname{GL}_m(\mathbb{Z})$. In this paper their square moment is evaluated and their bounds are proved in the case of consecutively dividing moduli. As an application, smooth sums of Fourier coefficients of a Maass form for $\operatorname{SL}_m(\mathbb{Z})$ against an exponential function $e(\alpha n)$ are estimated. These sums are proved to have rapid decay when $\alpha$ is a fixed rational number or a transcendental number with approximation exponent $\tau(\alpha) > m$. Non-trivial bounds are proved for these sums when $\tau(\alpha) > (m+1)/2$.

Voronoi summation formulae on GL(n)

We discover new Voronoi formulae for automorphic forms on GL(n) for n≥4. There are [n/2] different Voronoi formulae on GL(n), which are Poisson summation formulae weighted by Fourier coefficients of the automorphic form with twists by some hyper-Kloosterman sums.

Fourier coefficients of GL(N) automorphic forms in arithmetic progressions

We show that the multiple divisor functions of integers in invertible residue classes modulo a prime number, as well as the Fourier coefficients of GL(N) Maass cusp forms for all $${N \geq 2}$$ , satisfy a central limit theorem in a suitable range, generalizing the case N = 2 treated by Fouvry et al. (Commentarii Math Helvetici, 2014). Such universal Gaussian behaviour relies on a deep equidistribution result of products of hyper-Kloosterman sums.

On the Voronoĭ formula for GL( n )

We prove a general Vorono\u{\i} formula for cuspidal automorphic representations of ${\rm GL}(n)$ over number fields. This generalizes recent work by Miller-Schmid and Goldfeld-Li on Maass forms. Our method follows closely the adelic framework of integral representations of $L$-functions. The proof is flexible enough to allow ramification and we propose possible variants. For example the assumption that the additive twist is trivial at places where the representation is ramified is sufficient to obtain an explicit final statement with hyper-Kloosterman sums.

On unit root formulas for toric exponential sums

Starting from a classical generating series for Bessel functions due to Schlomilch, we use Dwork’s relative dual theory to broadly generalize unit-root results of Dwork on Kloosterman sums and Sperber on hyperkloosterman sums. In particular, we express the (unique) p-adic unit root of an arbitrary exponential sum on the torus Tn in terms of special values of the p-adic analytic continuation of a ratio of A-hypergeometric functions. In contrast with the earlier works, we use noncohomological methods and obtain results that are valid for arbitrary exponential sums without any hypothesis of nondegeneracy.

On the hyper-Kloosterman sum and its fourth power mean

The main purpose of this paper is to study the calculating problem of the fourth power mean of the hyper-Kloosterman sums, and give an exact calculating formula for them.

A NOTE ON LEHMER k-TUPLES

Let q > 2 be an odd number. For each integer a with 1 ≤ a ≤ q and (a, q) = 1, there exists one and only one [Formula: see text] with [Formula: see text] and [Formula: see text] such that [Formula: see text]. A Lehmer number is defined to be any integer a with [Formula: see text]. Let a = (a1, …, ak+1), b = (b1, …, bk+1) with 0 ≤ bi < ai for i = 1, …, k + 1, (a1 ⋯ ak+1, q) = 1, t = (t1, …, tk) satisfying 0 < t1, …, tk ≤ 1. Define [Formula: see text] The main purpose of this paper is to study the properties of N(a, b, t; q), and give a sharp asymptotic formula, by using the estimate for hyper-Kloosterman sums and properties of trigonometric sums.

Mean value of some exponential sums and applications to Kloosterman sums

Let q, m, n, k be integers with q ⩾ 3 and k ⩾ 1 , define the exponential sum S ( m , n , k ; q ) = ∑ ′ a = 1 q e ( m a + n a k q ) , where e ( x ) = e 2 π i x , ∑ ′ a denotes the summation over all a such that ( a , q ) = 1 . In this paper, we shall study the fourth power mean ∑ ′ n = 1 q | S ( m , n , k ; q ) | 4 , and give some identities on the mean value of classical Kloosterman sums and hyper-Kloosterman sums.

ON THE γ-TH HYPER-KLOOSTERMAN SUMS AND A PROBLEM OF D. H. LEHMER

For any integer k <TEX>$\geq$</TEX> 2, let P(c, k + 1;q) be the number of all k+1-tuples with positive integer coordinates (<TEX>$a_1,a_2,...,a_{k+1}$</TEX>) such that <TEX>$1{\leq}a_i{\leq}q$</TEX>, (<TEX>$a_i,q$</TEX>) = 1, <TEX>$a_1a_2...a_{k+1}{\equiv}$</TEX> c (mod q) and 2 <TEX>$\nmid$</TEX> (<TEX>$a_1+a_2+...+a_{k+1}$</TEX>), and E(c, k+1; q) = P(c, k+1;q) - <TEX>$\frac{{\phi}^k(q)}{2}$</TEX>. The main purpose of this paper is using the properties of Gauss sums, primitive characters and the mean value theorems of Dirichlet L-functions to study the hybrid mean value of the r-th hyper-Kloosterman sums Kl(h,k+1,r;q) and E(c,k+1;q), and give an interesting mean value formula.

POLYNOMIALS FOR HYPER-KLOOSTERMAN SUMS

Fix an integer m &gt; 1, and let [Formula: see text] denote the group of reduced residues modulo m and set ζm = exp (2 πi/m). Let [Formula: see text] denote the multiplicative inverse of x modulo m. The n-dimensional Kloosterman sums are defined by [Formula: see text] and satisfy the polynomial [Formula: see text] where the product is taken over a reduced system of residues modulo m. Here we give a natural factorization of fm(x); namely, [Formula: see text] where σ runs through the n + 1-st power classes modulo m. Questions concerning the determination of the factors [Formula: see text] (or at least their beginning coefficients), their reducibility over the rational field Q and duplications among the factors are studied. The treatment generalizes results of the author in the classical case n = 1, and relies heavily on the recent explicit evaluation of hyper-Kloosterman sums for prime powers.

A generalization on the difference between an integer and its inverse modulo Q

The main purpose of this paper is to use the properties of the Gauss sums, primitive characters and the mean value of Dirichlet L-functions to study the hybrid mean value of the error term E(n, 1, c, q) and the hyper-Kloosterman sums K(h, n+1, q), the asymptotic property of the mean square value Σc=1pE2(n, 1, c, p), and give two interesting mean value formulae.

Hybrid mean value results for a generalization on a problem of D.H. Lehmer and hyper-Kloosterman sums

The main purpose of this paper is by using the Fourier expansion for character sums and the mean value theorems of Dirichlet $L$-functions to give some hybrid mean value results for a generalization on a problem of D.H. Lehmer and hyper-Kloosterman sums.