Theorem 63.31.2. Given an eigenform $f$ with values in $\overline{\mathbf{Q}}_ l$ and eigenvalues $u_ v\in \overline{\mathbf{Z}}_ l^*$ then there exists

$\rho : \pi _1(X)\to \text{GL}_2(E)$

continuous, absolutely irreducible where $E$ is a finite extension of $\mathbf{Q}_\ell$ contained in $\overline{\mathbf{Q}}_ l$ such that $t_ v = \text{Tr}(\rho (F_ v))$, and $u_ v = q_ v^{-1}\det \left(\rho (F_ v)\right)$ for all places $v$.

Proof. See [D0]. $\square$

There are also:

• 2 comment(s) on Section 63.31: Automorphic forms and sheaves

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).