188 research outputs found
The Intersection of Two Fermat Hypersurfaces in P^3 via Computation of Quotient Curves
We study the intersection of two particular Fermat hypersurfaces in
over a finite field. Using the Kani-Rosen decomposition we study
arithmetic properties of this curve in terms of its quotients. Explicit
computation of the quotients is done using a Gr\"obner basis algorithm. We also
study the -rank, zeta function, and number of rational points, of the modulo
reduction of the curve. We show that the Jacobian of the genus 2 quotient
is -split
Curves of every genus with many points, I: Abelian and toric families
Let N_q(g) denote the maximal number of F_q-rational points on any curve of
genus g over the finite field F_q. Ihara (for square q) and Serre (for general
q) proved that limsup_{g-->infinity} N_q(g)/g > 0 for any fixed q. In their
proofs they constructed curves with many points in infinitely many genera;
however, their sequences of genera are somewhat sparse. In this paper, we prove
that lim_{g-->infinity} N_q(g) = infinity. More precisely, we use abelian
covers of P^1 to prove that liminf_{g-->infinity} N_q(g)/(g/log g) > 0, and we
use curves on toric surfaces to prove that liminf_{g-->infty} N_q(g)/g^{1/3} >
0; we also show that these results are the best possible that can be proved
with these families of curves.Comment: LaTeX, 20 page
Rational points on X_0^+ (p^r)
We show how the recent isogeny bounds due to \'E. Gaudron and G. R\'emond
allow to obtain the triviality of X_0^+ (p^r)(Q), for r>1 and p a prime
exceeding 2.10^{11}. This includes the case of the curves X_split (p). We then
prove, with the help of computer calculations, that the same holds true for p
in the range 10 < p < 10^{14}, p\neq 13. The combination of those results
completes the qualitative study of such sets of rational points undertook in
previous papers, with the exception of p=13.Comment: 16 pages, no figur
Ray class fields of global function fields with many rational places
A general type of ray class fields of global function fields is investigated.
The systematic computation of their genera leads to new examples of curves over
finite fields with comparatively many rational points.Comment: Latex2e, 27 pages, 20 tables, revised version as submitted to Acta
Arithmetic
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