Seminario t. de números

Title: Lower bounds for the maximal number of rational points of curves on finite fields

SPEAKER: Elisa Lorenzo García (Université de Neuchâtel y Université de Rennes 1)

DATE & TIME: Lunes 08 de enero de 2024 - 14:30

VENUE: Aula 420, Departamento de Matemáticas, UAM.

ABSTRACT: For a given genus $g>0$, we give lower bounds for the maximal number of rational points of an absolutely irreducible smooth projective curve of genus $g$ defined over the finite field $mathbb{F}_q$.

First, we give an explicit construction which produces curves of genus $g$ over $mathbb{F}_q$ with at least $1+q+4sqrt{q}-32$ points.

Then using sums of powers of traces of Frobenius of hyperelliptic curves, we obtain a lower bound of the form $1+q+1.71sqrt{q}$ valid for $g>2$ and $q>9$ odd.

Finally, and as a consequence of the Katz-Sarnak theory, we obtain for any given $g>0$, any $epsilon>0$ and all sufficiently large $q$, the existence of a curve of genus $g $ over $mathbb{F}_q$ with at least $1+q+(2g−epsilon)sqrt{q}$ rational points. In addition, we will go beyond this theory to try to explain the asymmetries observed in the distribution of the number of points and which are not detected by Katz-Sarnak.

This is a joint work with J. Bergström, E. Howe and C. Ritzenthaler.