On \(l\)-adic representations attached to Hilbert and Picard modular surfaces
Let \(L\) be a real quadratic number field and \(\pi\) a cohomological cuspidal automorphic representation of \(\mathrm{Res}_{L|\mathbb{Q}}\text{GL}(2)\) of weight 2. And let \(F\) be a totally real number field, \(E\) an imaginary quadratic extension of \(F\), and \(\Pi_f\) the finite component of a cohomological discrete series automorphic representation of \(\text{GU}(3)\) defined relative to \(E\mid F\). Under the assumption that \(F\) does not contain \(L\), the author computes the \(\ell\)-adic Lie algebra of the image of \(\Gamma_E=\mathrm{Gal}(\bar{\mathbb{Q}}| E)\) under the product \(\rho(\pi)|_{\Gamma_E}\times\phi(\Pi_f)\) of \(\ell\)-adic Galois representations associated with \(\pi\) and \(\Pi_f\) respectively. The author's approach is analogous to that of \textit{K. Ribet} [Math. Ann. 253, 43--62 (1980; Zbl 0421.14008)] for weight 2 eigenforms of \(\text{SL}(2)\).
scientific article; zbMATH DE number 5708638
scientific article