Computing the geometric endomorphism ring of a genus 2 Jacobian

TL;DR

This paper presents an algorithm based on Frobenius characteristic polynomials to compute the geometric endomorphism ring of genus 2 Jacobians, confirming existing database descriptions and exploring endomorphism fields.

Contribution

The paper introduces a new algorithm for computing endomorphism rings of genus 2 Jacobians using Frobenius properties, validating database data and analyzing endomorphism fields.

Findings

Algorithm successfully computes endomorphism rings for listed genus 2 curves.

Confirmed the accuracy of the LMFDB's descriptions of endomorphism structures.

Discussed methods to determine the fields over which endomorphisms are defined.

Abstract

We describe an algorithm, based on the properties of the characteristic polynomials of Frobenius, to compute $\operatorname{End}_{\overline{K}}(A)$ when $A$ is the Jacobian of a nice genus-2 curve over a number field $K$. We use this algorithm to confirm that the description of the structure of the geometric endomorphism ring of $\operatorname{Jac}(C)$ given in the LMFDB ($L$-functions and modular forms database) is correct for all the genus 2 curves $C$ currently listed in it. We also discuss the determination of the field of definition of the endomorphisms in some special cases.