Endomorphism algebras of geometrically split genus 2 Jacobians over Q

TL;DR

This paper classifies which geometric endomorphism algebras of split genus 2 Jacobians over Q can be realized by actual Jacobians over Q, providing explicit examples and non-examples.

Contribution

It demonstrates that 54 of the 92 possible endomorphism algebras are realized by Jacobians over Q, while 38 are not, with explicit examples for each case.

Findings

54 endomorphism algebras are realized by Jacobians over Q

38 endomorphism algebras do not correspond to Jacobians over Q

Explicit examples of both realizable and non-realizable cases

Abstract

The main result of [FG20] classifies the 92 geometric endomorphism algebras of geometrically split abelian surfaces defined over Q. We show that 54 of them arise as geometric endomorphism algebras of Jacobians of genus 2 curves defined over Q, and that the remaining 38 do not. In particular, we exhibit 38 examples of Qbar-isogeny classes of abelian surfaces defined over Q that do not contain any Jacobian of a genus 2 curve defined over Q, and for each of the 54 algebras that do arise we exhibit a curve realizing them.