Maps between curves and arithmetic obstructions

Abstract

Let X and Y be curves over a finite field. In this article we explore methods to determine whether there is a rational map from Y to X by considering L-functions of certain covers of X and Y and propose a specific family of covers to address the special case of determining when X and Y are isomorphic. We also discuss an application to factoring polynomials over finite fields.

Recovering algebraic curves from L-functions of Hilbert class fields

“…• C has no rational or quadratic Weierstrass points, • only one rational point of P 1 splits in C under π, • J C (F 3 ) has order 5. [SV,Example 4.1] provides two non-isomorphic curves satisfying these conditions. But these conditions will determine L(t, C, χ) for each character χ : J C (F 3 ) → C * , providing the desired example.…”

Using zeta functions to factor polynomials over finite fields

“…It even seems reasonable to guess that for a typical 1-parameter family of genus 4 curves, the family of Jacobians will satisfy the condition in Hypothesis Z. See [SV17] for some specific candidate families.…”