nLab Weil group

Not to be confused with Weyl group.

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Definition

Let $F$ be a p-adic field, with residue field denoted $\kappa$.

The Weil group $W_F$ is the subgroup of the Galois group $\mathrm{Gal}(\overline{F}/F)$ defined as the inverse image of Frobenius automorphisms $\mathrm{Frob}^{\mathbb{Z}}\subset \mathrm{Gal}(\overline{\kappa}/\kappa)$ under the surjective map $\mathrm{Gal}(\overline{F}/F)\to\mathrm{Gal}(\overline{\kappa}/\kappa)$.

Related entries

• Weil-Deligne representation

• local Langlands conjecture

References

• John Tate, Section 1 in: Number theoretic background, in: Automorphic forms, representations and L-functions, Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore. (1977), Part 2, Proc. Sympos. Pure Math., XXXIII, pages 3–26. Amer. Math. Soc., Providence, RI (ISBN:978-0-8218-3371-1, pdf, pdf)

See also:

• Wikipedia, Weil group