Examples of abelian surfaces failing the local-global principle for isogenies

TL;DR

This paper constructs examples of abelian surfaces over number fields that exhibit a failure of the local-global principle for isogenies, showing that local properties do not always imply global ones.

Contribution

It provides explicit examples of abelian surfaces over number fields and identifies modular abelian surfaces over  with similar failures, extending previous work.

Findings

Examples of abelian surfaces with local but not global ll-isogenies.

Identification of modular abelian surfaces over  with this property.

Extension of known phenomena to new classes of abelian surfaces.

Abstract

We provide examples of abelian surfaces over number fields $K$ whose reductions at almost all good primes possess an isogeny of prime degree $\ell$ rational over the residue field, but which themselves do not admit a $K$-rational $\ell$-isogeny. This builds on work of Cullinan and Sutherland. When $K=\mathbb{Q}$, we identify certain weight-$2$ newforms $f$ with quadratic Fourier coefficients whose associated modular abelian surfaces $A_f$ exhibit such a failure of a local-global principle for isogenies.