L-functions for GL(n)

Strassberg, Helen

doi:10.1007/bf01420427

Cited by 4 publications

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“…15), and then deduce (14. This problem is overcome (partially) in [12], in the context of more general representations of J(A), when A is a f ll matrix algebra. (This would not, of course, be sufficient in itself to prove (13.…”

Section: La P (S ψν)=π (2nr 2(ks~j} R(2(ks-j)}mentioning

confidence: 99%

Functional equations for L-functions of arithmetic orders.

1981

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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This paper is written s a continuation of [2], with consecutively numbered sections. Thus a reference to (n. m.), with n^9, means [2], (n. m.). Any unexplained notations are s in [2], but we shall avoid these wherever practicable. The bibliography is independent ofthatof[2].As before, A denotes a fmite-dimensional semisimple algebra over either the rational number field Q (global case) or some p-adic rational field Q p (local case) , and Λ is an order in A. Thus Λ is a Z-order in the global case, a Z p -order in the local case.The first part of the paper is concerned with the local case. Here we let M be a left ideal of A such that the index (Λ: M} is finite. Let ψ be a character (i.e., a continuous homomorphism to the unit circle in C) of A x which is of finite order and is trivial on the subgroup Aut^M). Then the local L-function L A (M, s, ψ} is defmed, s in (2. 1). More generally, for any character ψ of A* of finite order, and any function Φ Ε the space of Schwartz-Bruhat functions on A, we have the zeta-integral Ζ(Φ, s, ψ) = J Φ (χ) ψ(χ) \\x\\ s d* χ, Re(,s) > l, A*~ι as in (3. 2). Here, \\x\\ =(Λ:Λχ)~ι 9 and d x χ is a Haar measure on A* . The functions Ζ(Φ, s, ψ) admit analytic continuation, and satisfy a functional equation. This is basically well-known. We have already established the existence of analytic continuation in § 3, and we give a brief summary of the functional equation in § 10. This functional equation gives rise to a "local constant" or symmetry factor s A (s 9 ψ), which we here evaluate completely. This is done in two stages. We first give a formula for it (in § 10) in terms of a certain "non-abelian congruence Gauss sum" τ (ψ). This is a straightforward calculation based on the functional equation and the construction of the Standard L-function L A (s 9 ψ) in § 3. Later, in § 13, we evaluate T(I/^) in terms of classical Gauss sums. This is partly taken from unpublished joint work of the first author and A. Fr hlich.

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Section: La P (S ψν)=π (2nr 2(ks~j} R(2(ks-j)}mentioning

confidence: 99%