The Stacks project

Lemma 63.31.7. Switching $l$. Let $E$ be a number field. Start with

\[ \rho : \pi _1(X)\to SL_2(E_\lambda ) \]

absolutely irreducible continuous, where $\lambda $ is a place of $E$ not lying above $p$. Then for any second place $\lambda '$ of $E$ not lying above $p$ there exists a finite extension $E'_{\lambda '}$ and a absolutely irreducible continuous representation

\[ \rho ': \pi _1(X)\to SL_2(E'_{\lambda '}) \]

which is compatible with $\rho $ in the sense that the characteristic polynomials of all Frobenii are the same.

Proof. To prove the switching lemma use Theorem 63.31.3 to obtain $f\in C(\overline{\mathbf{Q}}_ l)$ eigenform ass. to $\rho $. Next, use Proposition 63.31.5 to see that we may choose $f\in C(E')$ with $E \subset E'$ finite. Next we may complete $E'$ to see that we get $f\in C(E'_{\lambda '})$ eigenform with $E'_{\lambda '}$ a finite extension of $E_{\lambda '}$. And finally we use Theorem 63.31.2 to obtain $\rho ': \pi _1(X) \to SL_2(E_{\lambda '}')$ abs. irred. and continuous after perhaps enlarging $E'_{\lambda '}$ a bit again. $\square$

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