Equidistribution of Gauss sums and Kloosterman sums

Let m be a positive integer. Fix a nontrivial additive character for each finite field F_q. To state the first result of this paper, we also fix r distinct multiplicative characters ₁,...,_r for each finite field F_q with more than r elements. We shall prove that when varies over multiplicative characters of F_q other than the m-th roots of the r-tuples of angles of Gauss sums are asymptotically equidistributed on the r-dimensional torus (S¹)^r as q goes to infinity.The n-dimensional Kloosterman sum over F_q at a F_q^× is One can define the angle (q,a) of Kl_n(q,a) in a suitable way. We shall prove that when a varies over nonzero elements of F_q, the q–1 angles (q,a^m) of Kloosterman sums are asymptotically equidistributed as q goes to infinity.Mathematics Subject Classification (2000) 11L05, 14F20     

Polynomial Gauss sums

A recent bound for exponential sums by Friedlander, Hansen and Shparlinski is extended to twisted exponential sums with general polynomial arguments. As a by-product a new result about perfect powers in certain products of polynomials is established.

     

Local units and Gauss sums

In this paper, we will determine the structure of a certain module which is related to the plus part of the ideal class groups in terms of the divisibility of Gauss sums in some local fields. This result is a generalization of a result of Iwasawa and the previous work of Ichimura and Hachimori.     

Incomplete higher-order Gauss sums

We consider the classical incomplete higher-order Gauss sums

     

Gauss sums and multinomial coefficients

Consider a Gauss sum for a finite field of characteristic p, where p is an odd prime. When such a sum (or a product of such sums) is a p-adic integer we show how it can be realized as a p-adic limit of a sequence of multinomial coefficients. As an application we generalize some congruences of Hahn and Lee to exhibit p-adic limit formulae, in terms of multinomial coefficients, for certain algebraic integers in imaginary quadratic fields related to the splitting of rational primes. We also give an example illustrating how such congruences arise from a p-integral formal group law attached to the p-adic unit part of a product of Gauss sums.     

Semi-local units modulo Gauss sums

For thep-th cyclotomic fieldk, Iwasawa proved thatp does not divide the class number of its maximal real subfield if and only if the odd part of the group of local units coincides with its subgroup generated by Jacobi sums related tok. We refine and give a quantitative version of this result for more general imaginary abelian fields. Our result is an analogy of the famous result on “semi-local units modulo cyclotomic units”. Partially supported by Grant-in-Aid for Scientific Research (C), Grant 09640054.     

Relative norms of Gauss sums

The explicit determination of the values of Gauss sums is a very classical problem and has some rather deep arithmetic consequences in classfield theory. Here we study the simpler problem of finding their relative norms. We give a complete determination of the relative norms of Gauss sums for norms whose values are known to be in an imaginary quadratic extension of the rational field.     

Dirichlet characters,Gauss sums and arithmetic Fourier transforms

In this paper, a general algorithm for the computation of the Fourier coefficients of 2π-periodic(continuous) functions is developed based on Dirichlet characters, Gauss sums and the generalized M¨obius transform. It permits the direct extraction of the Fourier cosine and sine coefficients. Three special cases of our algorithm are presented. A VLSI architecture is presented and the error estimates are given.     

Semi-local units modulo Gauss sums

For the p-th cyclotomic field k, Iwasawa proved that p does not divide the class number of its maximal real subfield if and only if the odd part of the group of local units coincides with its subgroup generated by Jacobi sums related to k. We refine and give a quantitative version of this result for more general imaginary abelian fields. Our result is an analogy of the famous result on “semi-local units modulo cyclotomic units”. Received: 2 May 1997 / Revised version: 11 November 1997     

An attack on Gauss,published by Legendre in 1820

The question of priority in the discovery of the method of least squares reached a climax when Legendre published an attack on Gauss in 1820. The background of the dispute is sketched, and this little known attack is presented in translation.     

Gauss sums and Kloosterman sums over residue rings of algebraic integers

Let denote the ring of integers of an algebraic number field of degree which is totally and tamely ramified at the prime . Write , where . We evaluate the twisted Kloosterman sum

where and denote trace and norm, and where is a Dirichlet character (mod ). This extends results of Salié for and of Yangbo Ye for prime dividing Our method is based upon our evaluation of the Gauss sum

which extends results of Mauclaire for .

     

Explicit multiplicative relations between Gauss sums

H. Hasse conjectured that all multiplicative relations between Gauss sums essentially follow from the Davenport-Hasse product formula and the norm relation for Gauss sums. While this is known to be false, very few counterexamples, now known as sign ambiguities, have been given. Here, we provide an explicit product formula giving an infinite class of new sign ambiguities and resolve the ambiguous sign in terms of the order of the ideal class of quadratic primes.     

Gauss sums over some matrix groups

In this note, we give explicit expressions of Gauss sums for general (resp. special) linear groups over finite fields, which involve classical Gauss sums (resp. Kloosterman sums). The key ingredient is averaging such sums over Borel subgroups, i.e., the groups of upper triangular matrices. As applications, we count the number of invertible matrices of zero-trace over finite fields and we also improve two bounds of Ferguson, Hoffman, Luca, Ostafe and Shparlinski in [R. Ferguson, C. Hoffman, F. Luca, A. Ostafe, I.E. Shparlinski, Some additive combinatorics problems in matrix rings, Rev. Mat. Complut. 23 (2010) 501–513].     

The triplication formula for Gauss sums

A new proof of the triplication formula for Gauss sums is given. It mimics an old proof of the analogous result for gamma functions. The techniques are formal and rely upon the character properties of fields. A new character sum evaluation is given.     

Bounds of Gauss sums in finite fields

We consider Gauss sums of the form

with a nontrivial additive character of a finite field of elements of characteristic . The classical bound becomes trivial for . We show that, combining some recent bounds of Heath-Brown and Konyagin with several bounds due to Deligne, Katz, and Li, one can obtain the bound on which is nontrivial for the values of of order up to . We also show that for almost all primes one can obtain a bound which is nontrivial for the values of of order up to .

     

Elliptic curve analogue of Legendre sequences

The Legendre symbol is applied to the rational points over an elliptic curve to output a family of binary sequences with strong pseudorandom properties. That is, both the well-distribution measure and the correlation measure of order k, which are evaluated by using estimation of certain character sums along elliptic curves, of the resulting binary sequences are “small”. A lower bound on the linear complexity profile of these sequences is also presented. Our results indicate that the behavior of such sequences is very similar to that of the Legendre sequences. Research partially supported by the Science and Technology Foundation of Putian City (No. 2005S04), the Open Funds of Key Lab of Fujian Province University Network Security and Cryptology (No. 07B005) and the Foundation of the Education Department of Fujian Province (No. JA07164). Author’s addresses: Department of Mathematics, Putian University, Putian, Fujian 351100, China; and Key Lab of Network Security and Cryptology, Fujian Normal University, Fuzhou, Fujian 350007, China