Examples of abelian surfaces failing the local-global principle for isogenies (Q1983151)

Let \(A\) be an abelian variety defined over a number field \(K\). The variety \(A\) is said to be \textbf{Hasse at \(\ell\)} if it does not admit a \(K\)-rational \(\ell\)-isogeny while all the reductions \(A_\mathfrak{P}\) at good primes \(\mathfrak{P}\) not dividing \(\ell\) do admit a \(K\)-rational \(\ell\)-isogeny. So it is a case where the local-global principle fails. The case of elliptic curves was studied by Sutherland and the two-dimensional case was initiated by \textit{J. Cullinan} [Int. J. Number Theory 8, No. 2, 321--334 (2012; Zbl 1276.11100)]. The paper gives examples of Hasse at \(\ell\) abelian surfaces over \(\mathbb{Q}\). Such an abelian variety \(A_f\) is obtained as the associated variety of a weight 2 newform \(f\) via the Eichler-Shimura construction. It is also proved (Theorem 1.3) that if \(A_f\) is a Hasse at \(\ell\) abelian surface, then \(f\) is congruent modulo \(\ell\) to a newform with complex multiplication.