A remark on the modularity of abelian varieties of \(GL_2\)-type over \(\mathbb{Q}\)
Let \(A/{\mathbb Q}\) be an abelian variety whose algebra of \(\mathbb Q\)-endomorphisms is a number field of degree equal to the dimension of \(A\) (\(A\) is said to be of \(GL_2\)-type over \({\mathbb Q}\)). The author proves that if \(A\) does not admit complex multiplication and \(A\) is ``modular over \(\mathbb C\)'', i.e., there exists a non-zero homomorphism \(J_1(N) \rightarrow A\) over \(\mathbb C\) (where \(J_1(N)\) is the Jacobian of the modular curve \(X_1(N)\) of level \(N\)), then \(A\) is ``modular over \(\mathbb Q\)'', i.e., isogenous over \(\mathbb Q\) to a factor of \(J_1(M)\) corresponding to a newform of weight two on \(\Gamma_1(M)\) for some integer \(M\). The proof is inspired by the case where \(A\) is one-dimensional [cf. \textit{B. Mazur}, Am. Math. Mon. 98, No. 7, 593-610 (1991; Zbl 0764.11021)].
scientific article; zbMATH DE number 1466188