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

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 enormous examine works at the automorphic L-functions connected through Langlands to reductive algebraic teams.

Chapter I makes a speciality of the research of Jacquet-Langlands equipment and the Einstein sequence and Langlands’ so-called “Euler products”. This bankruptcy explains how neighborhood and international 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 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 could be of worth to undergraduate and graduate arithmetic scholars.

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The proof that there exists a common denominator for the integrals Z(s, Wi, W2, fs), say for fs of type (ii), proceeds exactly as in Jacquet's theory. d. 8) reduces matters again to the case of type (ii). To see that Z(s, W\, W2, f 3 ) = 1 for appropriately chosen Wi and / , we fix f to be of type (ii) and such that its restriction to GL (O ) s 2 v has sup­ port in a compact subgroup fixing W\ and W2, with W{ the characteristic X (0V) . d. f for all "good sections" 2 s 8 of the integrals (of type (i), (ii) or (iii)).

Then A(S,Xv)fv) Ls(2s', Xv) vES or M(s)f = (IT L(2S~ vES IT 12 v¢S ,Xv )A(S,xv)fv) (IT 12) v¢s L(2s',X) · In order to establish the Proposition, it therefore suffices to prove the following Lemma. 4. For each v in S, and 1 Iv as above, L(2s', Xv) A(s, Xv)fv(g) is entire in s. · 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.

2) itl its place. - 59 - In [Shim2], yet another intriguing variation of the Rankin-Selberg identity is introduced, this time in order to extend Hecke's theory of Lfunctions for forms of integral weight to "symmetric squares" of these L-iunctions. More precisely, suppose once again that oo /(*)=Σα(η)β™* 2 n-l is a cusp form of {integral) weight k for SL2{%) and an eigenfunction of all the Hecke operators. 3) L\s, f) = Π ( ΐ - α 2 ρ - ) ( 1 - β Χ ρ - ) ( 1 - / 3 ? ρ - ) ' The goal of [Shim2] is to establish the analytic continuation and func­ 2 tional equation of L ( .

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