Choosing Embeddings $\overline{\mathbb{Q}}\hookrightarrow \overline{\mathbb{Q}}_p$

Choosing an embedding $\iota_p:\overline{\mathbb{Q}}\hookrightarrow \overline{\mathbb{Q}}_p$ is equivalent to choosing a prime of $\overline{\mathbb{Q}}$ above $p$.

This identifies a decomposition group in $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ with the local Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$, and hence fixes a notion of Frobenius at $p$.

Changing $\iota_p$ conjugates the resulting Frobenius and therefore conjugates the Satake parameter $\alpha_p$ in the $L$-group .

Consequence used in the letter: the Euler product $L(s)=\\prod_p L_p(s)$ changes only by finitely many local factors (the “bad primes”).