Modul: MAT076 Arbeitsgemeinschaft in Codierungstheorie und Kryptographie

## Lower bounds on the maximal number of rational points on curves over finite fields

Talk by Dr. Elisa Lorenzo Garcia

**Date:** 21.09.22 **Time:** 15.00 - 16.00 **Room:**

For a given genus $g\geq1$, we give lower bounds for the maximal number of rational points on a smooth projective absolutely irreducible curve of genus $g$ over $\mathbb{F}_q$. As a consequence of Katz-Sarnak theory, we first get for any given $g>0$, any $\epsilon>0$ and all $q$ large enough, the existence of a curve of genus $g$ over $\mathbb{F}_q$ with at least $1+q+(2g−\epsilon)\sqrt{q}$ rational points. Then using sums of powers of traces of Frobenius of hyperelliptic curves, we get a lower bound of the form $1+q+1.71\sqrt{q}$ valid for $g\geq3$ and odd $q\geq11$. Finally, explicit constructions of towers of curves improve this result, with a bound of the form $1+q+4\sqrt{q}−32$ valid for all $g\geq2$ and for all $q$.
This talk is based on a joint work with joint work with Jonas Bergström, Everett W. Howe, and Christophe Ritzenthaler.

(**This eSeminar will also be live-streamed on Zoom, using the same meeting details as previous seminars. If you do not have meeting details, please contact simran.tinani@math.uzh.ch **)