# Download e-book for iPad: Analytic Properties of Automorphic L-Functions by Stephen Gelbart, J. Coates, S. Helgason, Freydoon Shahidi

By Stephen Gelbart, J. Coates, S. Helgason, Freydoon Shahidi

ISBN-10: 0122791754

ISBN-13: 9780122791758

Analytic houses of Automorphic L-Functions is a three-chapter textual content that covers massive study works at the automorphic L-functions connected through Langlands to reductive algebraic teams.

Chapter I specializes in the research of Jacquet-Langlands tools and the Einstein sequence and Langlands’ so-called “Euler products. This bankruptcy explains how neighborhood and worldwide zeta-integrals are used to turn out the analytic continuation and practical equations of the automorphic L-functions hooked up to GL(2). bankruptcy II bargains with the advancements and refinements of the zeta-inetgrals for GL(n). bankruptcy III describes the implications for the L-functions L (s, ?, r), that are thought of within the consistent phrases of Einstein sequence for a few quasisplit reductive group.

This ebook should be of price to undergraduate and graduate arithmetic scholars.

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**Sample text**

In order to exploit the analytic properties of Z(s, φ) (and relate them to those of X(s,7r) = ΠΧ(θ, πν)) it is necessary to show that the X and ε factors which arise from the local versions of Z(s,(p) coincide with those defined in [Go J a] by way of the integrals 2(«β,Φ,/). This is a delicate enough technical problem which is further complicated by the fact that not all local representations have Whittaker models. Once settled, the converse theorem is easily established. Some uses of the converse theorem are reviewed in [Gel]; although the "trace formula" ostensibly renders the converse theorem redundant in establishing "Langlands functionality", the theorem remains a satisfying tool for recognizing automorphic cuspidal representations (cf.

F (0) th A(s,Xv) f L(2s' ,Xv)L(2s'+1,Xv) /-0 f If f v 180 P roo. , L(2s A(s,xv) f· I I t·· d f· f l ,xv) v IS C ear y en Ire In S, an we may now assume v IS 0 f-v. type (ii) or (iii). v,s(w( 1 Xl )g)dx . v,s(w( 1 havior of A(s, Xv)fv(s) depends only on the second integral. Note that the property of Iv being of type (i), (ii) or (iii) is preserved under right transla- tions by k in GL2(Ov), whereas bin B simply moves (across w( ~ :) and) out of the integral in question. Therefore we may assume 9 = bk with b = e = k.

E. which representations r) arise in this context, we need yet a few more definitions. For convenience, we reformulate the results of [Lai] from the slightly more general point of view of [Sha 1,3,8]. Hence G is not necessarily split, or semisimple, or of adjoint type. 2). Suppose that Η contains a Borel subgroup Β also defined over F , and write Β = T77, with Τ a maximal torus (also defined over F ) . Assume that P our maximal parabolic subgroup Ρ = GU contains B, and fix a special H H maximal compact subgroup K„ C Hv so that Η = BK = PK with H K = UK^.

### Analytic Properties of Automorphic L-Functions by Stephen Gelbart, J. Coates, S. Helgason, Freydoon Shahidi

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