64 research outputs found
On the distribution of the trace in the unitary symplectic group and the distribution of Frobenius
International audienceThe purpose of this article is to study the distribution of the trace on the unitary symplectic group. We recall its relevance to equidistribution results for the eigenvalues of the Frobenius in families of abelian varieties over finite fields, and to the limiting distribution of the number of points of curves. We give four expressions of the trace distribution if g = 2, in terms of special functions, and also an expression of the distribution of the trace in terms of elementary symmetric functions. In an appendix, we prove a formula for the trace of the exterior power of the identity representation
Number of points on abelian and Jacobian varieties over finite fields
We give upper and lower bounds on the number of points on abelian varieties
over finite fields, and lower bounds specific to Jacobian varieties. We also
determine exact formulas for the maximum and minimum number of points on
Jacobian surfaces.Comment: 28 page
Hypersurfaces in weighted projective spaces over finite fields with applications to coding theory
We consider the question of determining the maximum number of
-rational points that can lie on a hypersurface of a given degree
in a weighted projective space over the finite field , or in
other words, the maximum number of zeros that a weighted homogeneous polynomial
of a given degree can have in the corresponding weighted projective space over
. In the case of classical projective spaces, this question has
been answered by J.-P. Serre. In the case of weighted projective spaces, we
give some conjectures and partial results. Applications to coding theory are
included and an appendix providing a brief compendium of results about weighted
projective spaces is also included
Sur le nombre de points rationnels des variétés abéliennes et des Jacobiennes sur les corps finis
Hyperplane Sections of Grassmannians and the Number of MDS Linear Codes
AbstractWe obtain some effective lower and upper bounds for the number of (n,k)-MDS linear codes over Fq. As a consequence, one obtains an asymptotic formula for this number. These results also apply for the number of inequivalent representations over Fq of the uniform matroid or, alternately, the number of Fq-rational points of certain open strata of Grassmannians. The techniques used in the determination of bounds for the number of MDS codes are applied to deduce several geometric properties of certain sections of Grassmannians by coordinate hyperplanes
Toroidal automorphic forms, Waldspurger periods and double Dirichlet series
The space of toroidal automorphic forms was introduced by Zagier in the
1970s: a GL_2-automorphic form is toroidal if it has vanishing constant Fourier
coefficients along all embedded non-split tori. The interest in this space
stems (amongst others) from the fact that an Eisenstein series of weight s is
toroidal for a given torus precisely if s is a non-trivial zero of the zeta
function of the quadratic field corresponding to the torus.
In this paper, we study the structure of the space of toroidal automorphic
forms for an arbitrary number field F. We prove that it decomposes into a space
spanned by all derivatives up to order n-1 of an Eisenstein series of weight s
and class group character omega precisely if s is a zero of order n of the
L-series corresponding to omega at s, and a space consisting of exactly those
cusp forms the central value of whose L-series is zero.
The proofs are based on an identity of Hecke for toroidal integrals of
Eisenstein series and a result of Waldspurger about toroidal integrals of cusp
forms combined with non-vanishing results for twists of L-series proven by the
method of double Dirichlet series.Comment: 14 page
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