Counting curves over finite fields

TL;DR

This survey reviews recent advances in counting curves over finite fields, focusing on point counts, cohomology of local systems on moduli spaces, and implications for modular forms.

Contribution

It synthesizes recent results on the maximum number of points on curves and explores the cohomology of local systems, highlighting new connections to modular forms.

Findings

Maximum point counts on curves of low genus over finite fields

Cohomology of local systems on moduli spaces of curves

Implications for the theory of modular forms

Abstract

This is a survey on recent results on counting of curves over finite fields. It reviews various results on the maximum number of points on a curve of genus g over a finite field of cardinality q, but the main emphasis is on results on the Euler characteristic of the cohomology of local systems on moduli spaces of curves of low genus and its implications for modular forms.