Computing the geometric endomorphism ring of a genus 2 Jacobian

Abstract

We describe an algorithm, based on the properties of the characteristic polynomials of Frobenius, to compute ${\mathrm{End}}_{\overline{K}}(A)$ 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 $\mathrm{Jac}(C)$ 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.

Comments very welcome! v2. Updated to match published version

References in corpus (3)

• Rigorous computation of the endomorphism ring of a Jacobian
• A database of genus 2 curves over the rational numbers
• Real multiplication through explicit correspondences

Cited by in corpus (6)

• Rigorous computation of the endomorphism ring of a Jacobian
• From hyperelliptic to superelliptic curves
• A numerical transcendental method in algebraic geometry
• Explicit Kummer Theory for Elliptic Curves
• Bring's curve: Old and New
• Isogenous components of Jacobian surfaces