Genus bounds for curves with fixed Frobenius eigenvalues

TL;DR

This paper establishes explicit upper bounds on the genus of algebraic curves over finite fields, based on the Frobenius eigenvalues of their Jacobians, for varieties with specified Frobenius properties.

Contribution

It introduces a method to bound the genus of curves with Jacobians isogenous to products of given abelian varieties, based on Frobenius eigenvalues.

Findings

Derived explicit genus bounds for curves with prescribed Frobenius eigenvalues.

Provided a framework for analyzing Jacobians of curves over finite fields.

Extended understanding of the relationship between Frobenius eigenvalues and curve genus.

Abstract

For every finite collection C of abelian varieties over F_q, we produce an explicit upper bound on the genus of curves over F_q whose Jacobians are isogenous to a product of powers of elements of C.