Kloosterman sum identities and low-weight codewords in a cyclic code with two zeros

We apply relations of n-dimensional Kloosterman sums to exponential sums over finite fields to count the number of low-weight codewords in a cyclic code with two zeros. As a corollary we obtain a new proof for a result of Carlitz which relates one- and two-dimensional Kloosterman sums. In addition, we count some power sums of Kloosterman sums over certain subfields.     

Power-moments of SL3(ℤ) Kloosterman sums

Classical Kloosterman sums have a prominent role in the study of automorphic forms on GL₂ and further they have numerous applications in analytic number theory. In recent years, various problems in analytic theory of automorphic forms on GL₃ have been considered, in which analogous GL₃-Kloosterman sums (related to the corresponding Bruhat decomposition) appear. In this note we investigate the first four power-moments of the Kloosterman sums associated with the group SL₃(?). We give formulas for the first three moments and a nontrivial bound for the fourth.     

Codes associated with special linear groups and power moments of multi-dimensional Kloosterman sums

In this paper, we construct the binary linear codes C(SL(n, q)) associated with finite special linear groups SL(n, q), with both n,q powers of two. Then, via the Pless power moment identity and utilizing our previous result on the explicit expression of the Gauss sum for SL(n, q), we obtain a recursive formula for the power moments of multi- dimensional Kloosterman sums in terms of the frequencies of weights in C(SL(n, q)). In particular, when n = 2, this gives a recursive formula for the power moments of Kloosterman sums. We illustrate our results with some examples.     

3-Designs from the Z 4-Goethals Codes via a New Kloosterman Sum Identity

Recently, active research has been performed on constructing t-designs from linear codes over Z ₄. In this paper, we will construct a new simple 3 – (2 ^m , 7, 14/3 (2 ^m – 8)) design from codewords of Hamming weight 7 in the Z ₄-Goethals code for odd m 5. For 3 arbitrary positions, we will count the number of codewords of Hamming weight 7 whose support includes those 3 positions. This counting can be simplified by using the double-transitivity of the Goethals code and divided into small cases. It turns out interestingly that, in almost all cases, this count is related to the value of a Kloosterman sum. As a result, we can also prove a new Kloosterman sum identity while deriving the 3-design.     

Seventh power moments of Kloosterman sums

Evaluations of the n-th power moments S _n of Kloosterman sums are known only for n ⩽ 6. We present here substantial evidence for an evaluation of S ₇ in terms of Hecke eigenvalues for a weight 3 newform on Γ_O(525) with quartic nebentypus of conductor 105. We also prove some congruences modulo 3, 5 and 7 for the closely related quantity T ₇, where T _n is a sum of traces of n-th symmetric powers of the Kloosterman sheaf.     

Multiplicative character sums with the sum of g-ary Digits Function

We establish upper bounds for multiplicative character sums with the function σ_g (n) which computes the sum of the digits of n in a fixed base g ≥ 2. Our results may be viewed as analogues of some previously known results for exponential sums with sum of g-ary digits function. 2000 Mathematics Subject Classification Primary—11A63, 11B50, 11L40     

On Z4-Linear Goethals Codes and Kloosterman Sums

Studying the coset weight distributions of the Z₄-linear Goethals codes, e connect these codes with the Kloosterman sums. From one side, e obtain for some cases, of the cosets of weight four, the exact expressions for the number of code ords of weight four in terms of the Kloosterman sums. From the other side, e obtain some limitations for the possible values of the Kloosterman sums, hich improve the well known results due to Lachaud and Wolfmann kn:lac.     

The Linnik Conjecture. II

The Linnik conjecture is proved in the mean-square variant over the integer parameters (m,n) of the Kloosterman sum S(m,n;c). This mean may be called arithmetic, because the arithmetic of Kloosterman sums depends on the parameters (m,n). Bibliography: 10 titles.     

Some number-theoretic identities from group actions

We apply group actions to some natural situations like the natural ‘linear’ action of GL _r (Z _n ) and some of its subgroups to derive number-theoretic identities like

A New Proof of the Mullineux Conjecture

Let S _d denote the symmetric group on d letters. In 1979 Mullineux conjectured a combinatorial algorithm for calculating the effect of tensoring with an irreducible S _d-module with the one dimensional sign module when the ground field has positive characteristic. Kleshchev proved the Mullineux conjecture in 1996. In the present article we provide a new proof of the Mullineux conjecture which is entirely independent of Kleshchev's approach. Applying the representation theory of the supergroup GL(m | n) and the supergroup analogue of Schur-Weyl Duality it becomes straightforward to calculate the combinatorial effect of tensoring with the sign representation and, hence, to verify Mullineux's conjecture. Similar techniques also allow us to classify the irreducible polynomial representations of GL(m | n) of degree d for arbitrary m, n, and d.     

A hybrid mean value formula involving Kloosterman sums and Hurwitz zeta-function

The main purpose of this paper is using the mean value theorem of Dirichlet L-function and the estimates for character sums to study the asymptotic properties of a hybrid mean value of Kloosterman sums with the weight of Hurwitz zeta-function and the Cochrane sums, and give an interesting mean value formula for it.     

On sums of \mathit{SL}(3,\mathbb{Z}) Kloosterman sums

We show that sums of the $\mathit{SL}(3,\mathbb{Z})$ long element Kloosterman sum against a smooth weight function have cancelation due to the variation in argument of the Kloosterman sums, when each modulus is at least the square root of the other. Our main tool is Li’s generalization of the Kuznetsov formula on $\mathit{SL}(3,\mathbb{R})$ , which has to date been prohibitively difficult to apply. We first obtain analytic expressions for the weight functions on the Kloosterman sum side by converting them to Mellin–Barnes integral form. This allows us to relax the conditions on the test function and to produce a partial inversion formula suitable for studying sums of the long-element $\mathit{SL}(3,\mathbb{Z})$ Kloosterman sums.     

Trace Formula for p-Adics

We give a p-adic analogue of Selberg's trace formula relating the trace of a semigroup generated by a natural elliptic operator with a sum over contributions coming from closed geodesics. The construction uses probabilistic methods to define the generator.     

Trace formulae for differential pencils with spectral parameter dependent boundary conditions

For the eigenvalues λ_n of a differential operator, the series , generally speaking, diverges; however, it can be regularized by subtracting from λ_n the first terms of the asymptotic expansion, which interfere with the convergence of the series. The sum of such a regularized series is called the trace of Gelfand–Levitan type. A second‐order differential pencil on a finite interval with spectral parameter dependent boundary conditions is considered. We derive the regularized trace formulae of Gelfand–Levitan type for this operator. Copyright © 2013 John Wiley & Sons, Ltd.     

An Expansion Formula for q-Series and Applications

In this paper the author proves a q-expansion formula which utilizes the Leibniz formula for the q-differential operator. This expansion leads to new proofs of the Rogers–Fine identity, the nonterminating ₆₅ summation formula, and Watson's q-analog of Whipple's theorem. Andrews' identities for sums of three squares and sums of three triangular numbers are also derived. Other identities of Andrews and new identities for Hecke type series are also discussed.     

Homogeneity of Certain Invariant Distributions on the Lie Algebra of p-adic GLn

Let F be a non-Archimedean local field with ring of integers R and prime ideal . Suppose T is a GL _n (F)-invariant distribution on =M _n (F), the Lie algebra of GL _n (F). If T has support in the set of topologically nilpotent elements, then the restriction of T to the set of functions which are compactly supported and invariant under M _n ( ) may be expressed as a linear combination of nilpotent orbital integrals restricted to the same set of functions.     

Kloosterman identities over a quadratic extension II

We prove certain identities between Kloosterman integrals. They constitute the fundamental lemma of a relative trace formula for Hecke functions. The main application of the trace formula in question is the following result. Let E/F be a quadratic extension of number fields. A cuspidal automorphic representation of GL(n,EA) is distinguished by some unitary group if and only if it is the base change of an automorphic cuspidal representation of GL(n,FA).     

A relative Kuznietsov trace formula on G2

In this paper, we set up the general formulation to study distinguished residual representations of a reductive group G by the relative trace formula approach. This approach simplifies the argument of [JR], which deals with this type of relative trace formula for a special symmetric pair (GL(2n), Sp(2n)) and also works for non-symmetric, spherical pairs. To illustrate our idea and method, we complete our relative trace formula (both the geometric side identity and the spectral side identity) for the case (G ₂, SL(3)). Received: 6 February 1999     

A homological interpretation of Jantzen's sum formula

For a split reductive algebraic group, this paper observes a homological interpretation for Weyl module multiplicities in Jantzen's sum formula. This interpretation involves an Euler characteristic χ built from Ext groups between integral Weyl modules. The new interpretation makes transparent for GL_n (and conceivable for other classical groups) a certain invariance of Jantzen's sum formula under "Howe duality" in the sense of Adamovich and Rybnikov. For GL_n a simple and explicit general formula is derived for χ between an arbitrary pair of integral Weyl modules. In light of Brenti's work on certain R-polynomials, this formula raises interesting questions about the possibility of relating Ext groups between Weyl modules to Kazhdan-Lusztig combinatorics.     

On the 2<Emphasis Type="Italic">k</Emphasis>-th Power Mean of Inversion of <Emphasis Type="Italic">L</Emphasis>-functions with the Weight of the Gauss Sum

Abstract The main purpose of this paper is to use the estimate for character sums and the method of trigonometric sums to study the 2k-th power mean of the inversion of Dirichlet L-functions with the weight of the Gauss sums, and give a sharper asymptotic formula. This work is supported by the Doctorate Foundation of Xi’an Jiaotong University