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

Analytic houses of Automorphic L-Functions is a three-chapter textual content that covers enormous learn works at the automorphic L-functions hooked up by means of Langlands to reductive algebraic teams.

Chapter I makes a speciality of the research of Jacquet-Langlands tools 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 sensible equations of the automorphic L-functions connected 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 publication may be of price to undergraduate and graduate arithmetic scholars.

Show description

Read Online or Download Analytic Properties of Automorphic L-Functions PDF

Best mathematics_1 books

Educational Interfaces between Mathematics and Industry: Report on an ICMI-ICIAM-Study

This publication is the “Study publication” of ICMI-Study no. 20, which was once run in cooperation with the foreign Congress on and utilized arithmetic (ICIAM). The editors have been the co-chairs of the learn (Damlamian, Straesser) and the organiser of the research convention (Rodrigues). The textual content encompasses a accomplished file at the findings of the learn convention, unique plenary shows of the examine convention, studies at the operating teams and chosen papers from everywhere global.

Analytic Properties of Automorphic L-Functions

Analytic houses of Automorphic L-Functions is a three-chapter textual content that covers huge learn works at the automorphic L-functions hooked up by means of Langlands to reductive algebraic teams. bankruptcy I makes a speciality of the research of Jacquet-Langlands tools and the Einstein sequence and Langlands’ so-called “Euler products”.

Extra info for Analytic Properties of Automorphic L-Functions

Example text

50 - In a sense which will become clearer in the next paragraph, Jacquet's choice of sections s —• ff,x(d) already has built into it the "normalizing factor" L(2s,x) which globally knocks out (infinitely many) unwanted poles of E(g,s), and locally produces the correct Langlands-Euler factor. ( A . 2 ) A n A l t e r n a t e Approach t o [Jacquet] Our purpose is to redo Jacquet's theory without recourse to Fourier transforms or the use of Schwartz-Bruhat functions in denning permissible sections s — • fs{g)We begin with the Eisenstein series E(s,g,fa)= Σ fa(yg), BF\Gp 1 1 5 12 1 δ t eh where fs £ JB.

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. For simplicity, let us suppose v is a finite place. Since invariant by [ 1_1 -x 0 ] for 1 X] [ 1 Ixl Iv is then right large, the Iwasawa decomposition 1_1 -x 0] = [ x-I -1 ] lOx - 53 - implies that the second integral above (with 9 = e) equals Thus we need only examine the analytic nature of the expression If fv is of type (ii) above, then fv,s(e) is independent of s, and 1~;~~~:) is an entire function of s, whether or not Xv is unramified (since the integral ~:r:1 small Xv(x )lxI 2s'-ldx has the same meromorphic behavior as L(2s', Xv».

Has the matrix form g\ + ig2 with respect to some basis 3 &1>&2>&3 of if , with <7i,02 in GL^(F), then it is easy to check that g - 27 - (regarded as a linear transformation of the six dimensional space if 3 over F) has the matrix form with respect to the basis ( 92 i 92 2 \ \ 92 91 ) &2, &3, i&2, ^ 3 } . e. , G is indeed defined over F. On the other hand, as a group over K, G « GL$ via the isomorphism 2 / 91 i 92 λ \9292 91 J 91 + 192 L Thus G° is just GL3( (C). 1 0 1 0 1 • 0 0. 1 0 1 0 where 1• 0 0.

Download PDF sample

Rated 4.56 of 5 – based on 24 votes
 

Author: admin