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

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