76 research outputs found
On a conjecture of Helleseth
We are concern about a conjecture proposed in the middle of the seventies by
Hellesseth in the framework of maximal sequences and theirs cross-correlations.
The conjecture claims the existence of a zero outphase Fourier coefficient. We
give some divisibility properties in this direction
Kloosterman sums, elliptic curves, and irreducible polynomials with prescribed trace and norm
Let \F_q () be a finite field. In this paper the number of
irreducible polynomials of degree in \F_q[x] with prescribed trace and
norm coefficients is calculated in certain special cases and a general bound
for that number is obtained improving the bound by Wan if is small compared
to . As a corollary, sharp bounds are obtained for the number of elements in
\F_{q^3} with prescribed trace and norm over \F_q improving the estimates
by Katz in this special case. Moreover, a characterization of Kloosterman sums
over \F_{2^r} divisible by three is given generalizing the earlier result by
Charpin, Helleseth, and Zinoviev obtained only in the case odd. Finally, a
new simple proof for the value distribution of a Kloosterman sum over the field
\F_{3^r}, first proved by Katz and Livne, is given.Comment: 21 pages; revised version with somewhat more clearer proofs; to
appear in Acta Arithmetic
Proof of a Conjectured Three-Valued Family of Weil Sums of Binomials
We consider Weil sums of binomials of the form , where is a finite field, is
the canonical additive character, , and .
If we fix and and examine the values of as runs
through , we always obtain at least three distinct values unless
is degenerate (a power of the characteristic of modulo ).
Choices of and for which we obtain only three values are quite rare and
desirable in a wide variety of applications. We show that if is a field of
order with odd, and with , then
assumes only the three values and . This
proves the 2001 conjecture of Dobbertin, Helleseth, Kumar, and Martinsen. The
proof employs diverse methods involving trilinear forms, counting points on
curves via multiplicative character sums, divisibility properties of Gauss
sums, and graph theory.Comment: 19 page
The Divisibility Modulo 4 of Kloosterman Sums over Finite Fields of Characteristic 3
Recently Garashuk and Lisonek evaluated Kloosterman sums
K (a) modulo 4 over a finite field F3m in the case of even K (a). They posed it as an open
problem to characterize elements a in F3m for which K (a) ≡ 1 (mod4) and K (a) ≡ 3 (mod4). In
this paper, we will give an answer to this problem. The result allows us to count the number of
elements a in F3m belonging to each of these two classes
On the Distribution of the Number of Points on Algebraic Curves in Extensions of Finite Fields
Let \cC be a smooth absolutely irreducible curve of genus defined
over \F_q, the finite field of elements. Let # \cC(\F_{q^n}) be the
number of \F_{q^n}-rational points on \cC. Under a certain multiplicative
independence condition on the roots of the zeta-function of \cC, we derive an
asymptotic formula for the number of such that (# \cC(\F_{q^n})
- q^n -1)/2gq^{n/2} belongs to a given interval \cI \subseteq [-1,1]. This
can be considered as an analogue of the Sato-Tate distribution which covers the
case when the curve \E is defined over \Q and considered modulo consecutive
primes , although in our scenario the distribution function is different.
The above multiplicative independence condition has, recently, been considered
by E. Kowalski in statistical settings. It is trivially satisfied for ordinary
elliptic curves and we also establish it for a natural family of curves of
genus .Comment: 14 page
Artin Conjecture for p-adic Galois Representations of Function Fields
For a global function field K of positive characteristic p, we show that
Artin conjecture for L-functions of geometric p-adic Galois representations of
K is true in a non-trivial p-adic disk but is false in the full p-adic plane.
In particular, we prove the non-rationality of the geometric unit root
L-functions.Comment: Remove the condition 6|k in Lemma 3.8; final versio
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