In his 1975 Inventiones paper "On $\ell$-adic representations attached to modular forms", Ken Ribet shows that if $f_1, f_2$ are any two cuspidal modular eigenforms for $\operatorname{SL}_2(\mathbb{Z})$, not Galois-conjugate to each other, then the image of the Galois representation on $\rho_{f_1, \ell} \times \rho_{f_2, \ell}$ is "as large as possible" for all but finitely many $\ell$; i.e the image is $$ \{ (u, v) \in \operatorname{GL}_2(\mathcal{O}_{f_1, \ell}) \times \operatorname{GL}_2(\mathcal{O}_{f_2, \ell}) : \exists x \in \mathbb{Z}_\ell^\times \text{ such that } \det(u) = x^{k_1 - 1}, \det(v) = x^{k_2 - 1}\} $$ where $\mathcal{O}_{f_1, \ell}$ and $\mathcal{O}_{f_2, \ell}$ are the rings of integers of the relevant completions of the coefficient fields of $f_1$ and $f_2$, and $k_1, k_2$ are the weights. Moreover, for any prime $\ell$ the image is open in the above group.

Has this theorem been generalized to modular forms of higher levels and non-trivial characters? If so, what is a good reference for this? (I know lots of references for "big image" statements for individual modular forms, but for pairs of modular forms I don't know where to look.)


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.