Numerical computation of endomorphism rings

TL;DR

This paper introduces practical numerical techniques for computing the period matrix of algebraic curves, enabling heuristic calculations of automorphisms, isomorphisms, Jacobian decompositions, and applications like Prym varieties.

Contribution

It provides novel numerical methods for analyzing algebraic curves and their Jacobians, including automorphisms and isogeny decompositions, with applications to morphisms and Prym varieties.

Findings

Effective numerical algorithms for period matrix computation

Heuristic methods for automorphism and isomorphism detection

Applications to Prym variety identification

Abstract

We give practical numerical methods to compute the period matrix of a plane algebraic curve (not necessarily smooth). We show how automorphisms and isomorphisms of such curves, as well as the decomposition of their Jacobians up to isogeny, can be calculated heuristically. Particular applications include the determination of (generically) non-Galois morphisms between curves and the identification of Prym varieties.