Choosing an embedding $\overline{\mathbb Q}\hookrightarrow\overline{\mathbb Q}_p$

Choosing an embedding

\iota_p:\overline{\mathbb Q} \hookrightarrow \overline{\mathbb Q}_p

extending $\mathbb Q\hookrightarrow\mathbb Q_p$ is equivalent to choosing a place of $\overline{\mathbb Q}$ above $p$ . It identifies the decomposition subgroup at that place with

\operatorname{Gal}(\overline{\mathbb Q}_p/\mathbb Q_p),

up to the usual conjugacy in the absolute Galois group $\operatorname{Gal}(\overline{\mathbb Q}/\mathbb Q)$ .

Effect of changing the embedding

A different choice gives a conjugate decomposition subgroup. At an unramified prime, the corresponding inertia subgroup acts trivially, and the Frobenius elements are conjugate. Therefore conjugacy-invariant data such as characteristic polynomials, Satake conjugacy classes, and local $L$ -factors do not change.

This choice should not be confused with omitting finitely many ramified or bad places from an Euler product; that finite set is determined by the global arithmetic data, not by changing $\iota_p$ .

General number-field form

For a number field $F$ and a finite place $v$ , an embedding $\overline F\hookrightarrow\overline{F_v}$ extending $F\hookrightarrow F_v$ selects a decomposition subgroup $\Gamma_{F_v}\subset\Gamma_F$ . It is needed to formulate the localization of a global Galois representation and hence local–global compatibility, while the final local conjugacy class is independent of the choice.

Coefficient embeddings are separate

Comparing a complex automorphic parameter with an $\ell$ -adic Galois representation also requires an isomorphism $\iota:\overline{\mathbb Q}_\ell\simeq\mathbb C$ . That coefficient-field choice is logically distinct from selecting a place above $p$ .

References

Jean-Pierre Serre, Local Fields, Springer, 1979, Chapters I and IV.