Decomposing Jacobians of Curves over Finite Fields in the Absence of Algebraic Structure

TL;DR

This paper investigates conditions under which the L-polynomial of one curve divides that of another over finite fields, linking point counts over extensions to polynomial divisibility and exploring implications for exponential sums.

Contribution

It establishes a new theorem connecting point count behavior over extensions to L-polynomial divisibility and introduces a conjecture relating L-polynomials to exponential sums.

Findings

Divisibility of L-polynomials follows from point count conditions and an additional assumption.

Application to a family of curves related to exponential sum conjecture.

Proposes a new conjecture on L-polynomials equivalent to the exponential sums conjecture.

Abstract

We consider the issue of when the L-polynomial of one curve over $\F_q$ divides the L-polynomial of another curve. We prove a theorem which shows that divisibility follows from a hypothesis that two curves have the same number of points over infinitely many extensions of a certain type, and one other assumption. We also present an application to a family of curves arising from a conjecture about exponential sums. We make our own conjecture about L-polynomials, and prove that this is equivalent to the exponential sums conjecture.