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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
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