## Galois $$\epsilon$$-factors modulo roots of unity.(English)Zbl 0557.12011

Let F be a locally compact non-Archimedean field, $$\bar F$$ a separable algebraic closure of F, $$G_ F$$ the Galois group of $$\bar F$$ over F, and p the residue characteristic of F. We fix a non-trivial character $$\psi$$ of the additive group F, and on this group we take the Haar measure dx which is self-dual for $$\psi$$. For every continuous finite-dimensional complex representation $$\sigma$$ of the group $$G_ F$$, Langlands and Deligne have proved the existence of the local constant of functional equations of L-functions attached to $$\sigma$$,$$\psi$$ and dx, also known as the $$\epsilon$$-factor of $$\sigma$$. We consider $$\epsilon$$ ($$\sigma$$,$$\psi$$,dx) as a function $$\epsilon$$ ($$\sigma)$$ of the complex parameter s and write $$\epsilon$$ ’($$\sigma)$$ for its value at $$s=1/2$$. In view of the local Langlands conjecture it is of foremost importance to understand more about the $$\epsilon$$-factors. P. Deligne and the author have investigated the $$\epsilon$$-factor $$\epsilon$$ ($$\sigma)$$ when $$\sigma$$ is twisted by another less ramified representation [ibid. 64, 89-118 (1981; Zbl 0442.12012)]. In particular they showed that this behaviour is remarkably simple if one considers the values of $$\epsilon$$ ($$\sigma)$$ only modulo the group $$\mu$$ of roots of unity in $${\mathbb{C}}^{\times}$$ of p-power order.
In the present paper an easy formula (involving a quadratic Gauss sum) is given for $$\epsilon$$ ’($$\sigma)$$ modulo $$\mu$$, in a number of interesting cases, notably when $$\sigma$$ is nonminimal, or primitive, or irreducible of degree p with Artin exponent prime to p. This plays a crucial role in the proof of the Langlands correspondence for GL(p,F) (p odd) by Kutzko, Moy and the author.

### MSC:

 11S37 Langlands-Weil conjectures, nonabelian class field theory 22E50 Representations of Lie and linear algebraic groups over local fields

### Citations:

Zbl 0451.12006; Zbl 0442.12012
Full Text:

### References:

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