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Non-Abelian congruence Gauss sums and p-adic simple algebras. (English) Zbl 0558.12007

This paper develops a detailed theory of Gauss sums for central simple algebras A (of finite degree) of a finite extension F of \({\mathbb{Q}}_ p\). The case where \(A=F\) is the well known theory of Abelian congruence Gauss sums, and the authors’ previous paper [Gauss sums and p-adic division algebras (Lect. Notes Math. 987) (1983; Zbl 0507.12008)] dealt with the case of division algebras over F. Here the general case is defined and thoroughly studied. For this the authors first introduce after Benz, the so-called principal orders of A; the normalizers G(\({\mathfrak A})\), in \(A^{\times}\), of principal orders \({\mathfrak A}\) of A are precisely the maximal compact-mod-center subgroups of \(A^{\times}.\)
Let us fix a non-trivial additive character of F. Then to any irreducible admissible representation \(\rho\) of G(\({\mathfrak A})\) (over an algebraically closed field B of characteristic not p) is attached a Gauss sum \(\tau\) (\(\rho)\). The absolute value of this Gauss sum (when \(B={\mathbb{C}})\) is computed and a functional equation between \(\tau\) (\(\rho)\) and \(\tau (\rho^ v)\) is proved; reduction modulo a prime number \(\ell \neq p\) is also investigated, as well as the behaviour of \(\tau\) (\(\rho)\) under Galois actions. Finally the relationship of those Gauss sums with local constants (\(\epsilon\)-factors) for supercuspidal admissible irreducible representations of \(A^{\times}\) is studied, leading to the conjecture that any such representation of \(A^{\times}\) contains some ”non- degenerate” representation of some G(\({\mathfrak A})\) as above. Throughout the paper the parallelism with Galois Gauss sums is stressed.
Reviewer: G.Henniart

MSC:

11S45 Algebras and orders, and their zeta functions
22E50 Representations of Lie and linear algebraic groups over local fields
16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
11L10 Jacobsthal and Brewer sums; other complete character sums

Citations:

Zbl 0507.12008
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