Unramified prime and Frobenius element

Let $K/F$ be a finite Galois extension of global fields, let $v$ be a place of $F$ unramified in $K$ , and choose $w\mid v$ . The inertia subgroup is then trivial, and reduction gives an isomorphism from the decomposition group:

D_w \xrightarrow{\sim} \operatorname{Gal}(\kappa(w)/\kappa(v)).

The arithmetic Frobenius $\operatorname{Frob}^{\mathrm{arith}}_w$ is the element acting on the residue field by

x\longmapsto x^{q_v}.

Its inverse is the geometric Frobenius $\operatorname{Frob}^{\mathrm{geom}}_w$ .

Conjugacy class

Changing $w$ conjugates the decomposition subgroup and its Frobenius. Consequently the conjugacy class $\operatorname{Frob}_v\subset\operatorname{Gal}(K/F)$ is independent of the chosen place above $v$ .

For an infinite Galois extension or an $\ell$ -adic representation, Frobenius is recorded in the appropriate finite quotient or as a conjugacy class modulo the inertia subgroup.

Weil-group convention

For a nonarchimedean local field, the Weil group maps onto $\mathbb Z$ , generated by a chosen arithmetic or geometric Frobenius. Local class field theory likewise has two reciprocal normalizations: a uniformizer can map to arithmetic Frobenius or its inverse.

Langlands role

Unramified Galois parameters and Satake parameters are compared at Frobenius. Changing convention inverts the Frobenius element and its eigenvalues, so the local $L$ -factor formula must change with it.

Relation to the letter

The letter's element $\sigma$ is the Galois component of its unramified class $\alpha_p$ . Reading its determinant formula requires fixing which of the two Frobenius conventions $\sigma$ denotes.

References

Jean-Pierre Serre, Local Fields, Springer, 1979.
John Tate, “Number theoretic background,” Proc. Sympos. Pure Math. 33, part 2, 1979.