Divisibility of L-Polynomials for a Family of Curves

TL;DR

This paper investigates the divisibility properties of L-polynomials among algebraic curves, proving one of two conjectures related to their divisibility and Jacobian structures.

Contribution

It proves one of the two conjectures about L-polynomial divisibility for a specific family of curves, advancing understanding of their algebraic and arithmetic properties.

Findings

Proved one conjecture on L-polynomial divisibility

Established conditions relating curves and their Jacobians

Enhanced understanding of algebraic curve invariants

Abstract

We consider the question of when the L-polynomial of one curve divides the L-polynomial of another curve. A theorem of Tate gives an answer in terms of jacobians. We consider the question in terms of the curves. The last author gave an invited talk at the 12th International Conference on Finite Fields and Their Applications on this topic, and stated two conjectures. In this article we prove one of those conjectures.