# Download e-book for kindle: 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 homes of Automorphic L-Functions is a three-chapter textual content that covers substantial learn works at the automorphic L-functions hooked up by way of 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 end up the analytic continuation and useful 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 consequences 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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**Extra resources for Analytic Properties of Automorphic L-Functions**

**Sample text**

5, and will of course playa crucial role in the theory of L-functions. 5) which expresses M(s,1I'") in terms of certain automorphic L-functions. Let L P denote the parabolic subgroup of L H corresponding to P in H, and let L7J denote the Lie algebra of the unipotent radical of L P. The group LG acts on LTJ via the adjoint action, and the corresponding representation :L G ~ GL(LTJ) decomposes into the sum of certain "eigenvalue" rep- T resentations Ti, i = 1, ... , m. For P = B C H = SL2, this representation T is itself already irreducible; in particular, m = 1 and T = Tt.

Jacquet] is a 1972 sequel to [JL], which analyzes the L-functions L(s,w\ χ π2) attached to pairs of automorphic cuspidal representations of GL(2); in this work, the classical papers of [Rankinl] and [Selberg] play the same role as Hecke's work on the Dirichlet series L(s, / ) plays in [JL]. Thus the method developed in [Jacquet] is often referred to as the "Rankin-Selberg method". Classically, the goal is to study analytic properties of the Dirichlet series oo n=l when Σ™=ι αηε 2πίηζ and Σ™=ι Ke define automorphic forms of 2ninz weight k for SL2{TL).

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^. Let A denote the maximal split torus in the center of G, with real Lie algebra U = Hom(X(G)F,JR) = Hom(X(A)F,JR) . ) consider the homomorphism Hp : G& —• U defined by exp(Hp(g)(X)) = - 81 - l[\x(g )\ v v Now where X E X(G)p, and g = (gv) E GA. We extend Hp to HA by making it trivial on U[ and Kf. 1) Let H { [~ a~l] } = A. , those generating U (the unipotent radical of B), and ~ C ep+ the simple roots.

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

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