Theorem 64.31.2 (03VT)—The Stacks project

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Part 3: Topics in Scheme Theory
Chapter 64: The Trace Formula
Section 64.31: Automorphic forms and sheaves
Theorem 64.31.2 (cite)

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Theorem 64.31.2. 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]. $\square$

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