676 research outputs found
A note on the moments of Kloosterman sums
In this note, we deduce an asymptotic formula for even power moments of
Kloosterman sums based on the important work of N. M. Katz on Kloosterman
sheaves. In a similar manner, we can also obtain an upper bound for odd power
moments. Moreover, we shall give an asymptotic formula for odd power moments of
absolute Kloosterman sums. Consequently, we find that there are infinitely many
such that as Comment: 8 pages, to appear in PAM
Recursive formulas generating power moments of multi-dimensional Kloosterman sums and -multiple power moments of Kloosterman sums
In this paper, we construct two binary linear codes associated with
multi-dimensional and multiple power Kloosterman sums (for any fixed )
over the finite field . Here is a power of two. The former
codes are dual to a subcode of the binary hyper-Kloosterman code. Then we
obtain two recursive formulas for the power moments of multi-dimensional
Kloosterman sums and for the -multiple power moments of Kloosterman sums in
terms of the frequencies of weights in the respective codes. This is done via
Pless power moment identity and yields, in the case of power moments of
multi-dimensional Kloosterman sums, much simpler recursive formulas than those
associated with finite special linear groups obtained previously.Comment: 14 page
A Recursive Formula for Power Moments of 2-Dimensional Kloosterman Sums Assiciated with General Linear Groups
In this paper, we construct a binary linear code connected with the
Kloosterman sum for . Here is a power of two. Then we obtain a
recursive formula generating the power moments 2-dimensional Kloosterman sum,
equivalently that generating the even power moments of Kloosterman sum in terms
of the frequencies of weights in the code. This is done via Pless power moment
identity and by utilizing the explicit expression of the Kloosterman sum for
.Comment: 9 page
Codes Associated with Orthogonal Groups and Power Moments of Kloosterman Sums
In this paper, we construct three binary linear codes ,
, , respectively associated with the orthogonal
groups , , , with powers of two. Then
we obtain recursive formulas for the power moments of Kloosterman and
2-dimensional Kloosterman sums in terms of the frequencies of weights in the
codes. This is done via Pless power moment identity and by utilizing the
explicit expressions of Gauss sums for the orthogonal groups. We emphasize
that, when the recursive formulas for the power moments of Kloosterman sums are
compared, the present one is computationally more effective than the previous
one constructed from the special linear group . We illustrate our
results with some examples
Codes Associated with and Power Moments of Kloosterman Sums with Trace One Arguments
We construct a binary linear code , associated with the orthogonal
group . Here is a power of two. Then we obtain a recursive formula
for the odd power moments of Kloosterman sums with trace one arguments in terms
of the frequencies of weights in the codes and . This
is done via Pless power moment identity and by utilizing the explicit
expressions of Gauss sums for the orthogonal groups
Ternary Codes Associated with O^-(2n,q) and Power Moments of Kloosterman Sums with Square Arguments
In this paper, we construct three ternary linear codes associated with the
orthogonal group O^-(2,q) and the special orthogonal groups SO^-(2,q) and
SO^-(4,q). Here q is a power of three. Then we obtain recursive formulas for
the power moments of Kloosterman sums with square arguments and for the even
power moments of those in terms of the frequencies of weights in the codes.
This is done via Pless power moment identity and by utilizing the explicit
expressions of "Gauss sums" for the orthogonal and special orthogonal groups
O^-(2n,q) and SO^-(2n,q).Comment: 16 page
Ternary codes associated with symplectic groups and power moments of Kloosterman sums with square arguments
In this paper, we construct two ternary linear codes associated with the
symplectic groups Sp(2,q) and Sp(4,q). Here q is a power of three. Then we
obtain recursive formulas for the power moments of Kloosterman sums with square
arguments and for the even power moments of those in terms of the frequencies
of weights in the codes. This is done via Pless power moment identity and by
utilizing the explicit expressions of "Gauss sums" for the symplectic groups
Sp(2n,q).Comment: No comment
Power moments of Kloosterman sums
In this paper we give an essential treatment for estimating arbitrary
integral power moments of Kloosterman sums over the residue class ring. For
prime moduli we derive explicit estimates, and for prime-power moduli we prove
concrete formulas using computations with Igusa zeta functions.Comment: To appear in Journal of Number Theory, 25 page
Supercharacters, elliptic curves, and the sixth moment of Kloosterman sums
We connect the sixth power moment of Kloosterman sums to elliptic curves.
This yields an elementary proof that with are .Comment: 13 page
Level Reciprocity in the twisted second moment of Rankin-Selberg L-functions
We prove an exact formula for the second moment of Rankin-Selberg
-functions twisted by , where is a
fixed holomorphic cusp form and is summed over automorphic forms of a given
level . The formula is a reciprocity relation that exchanges the twist
parameter and the level . The method involves the Bruggeman/Kuznetsov
trace formula on both ends; finally the reciprocity relation is established by
an identity of sums of Kloosterman sums.Comment: 13 page
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