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

Abstract

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