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 connected 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 international zeta-integrals are used to turn out the analytic continuation and practical 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 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 could be of price to undergraduate and graduate arithmetic scholars.

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**Additional resources for Analytic Properties of Automorphic L-Functions**

**Example text**

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)). We also let L ( l — θ, πι χ 7Γ2) denote the corresponding generator of the ideal W\, W , A*(s)f). Z(l — θ, W[, W\ and f\ such that 2 By definition, there exists a finite number of Ν Σ Z(S, Wl Wi ft) = L(S, TT!

Thus we get the desired functional equation L(s, πχ χ 7 τ 2) = e(s, πχ χ π 2 ) £ ( 1 - s, πχ χ π 2 ) with "monomial" factor ε. The analytic continuation of L(s, πχ χ π 2 ) , on the other hand, follows from the identity Ι(θ,7Γΐ x π 2 ) = ^ipi{g)ip 2{g)E{g,s,f s)dg . In particular, according to what we just learned about the poles of E(g, s, fs) (for these good / = Π / υ ) , we know that the only possible poles of L(s, πχ χ f π 2 ) are those of L(2s ,x). This result is consistent with Jacquet's asser tion that the only possible poles occur when | | When χ is not of the form | 2 _ 2s = χ or | | 2 s = χ - 1 .

3) [ 3 χ Φ(χ)χ(χ)\χ\ ά χ JF analyzed in [Tatel]. The method of [Go Ja] (and Section 14 of [JL]) is thus a natural generalization of the method used by Tate to establish the analytic continuation and functional equation of Hecke's L-series Χ(θ,χ). The idea of this generalization was apparently first (systematically) de veloped by Tamagawa and Godement in the 1960's, but the method is still often referred to as "Tate's method" (even in the general context of Mn(D)). In general, suppose (π, V) is an unramified representation of Φ(χ) is equal to the characteristic function of Mn(Ojr), {n(g)vo,vo) with VQ (resp.

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

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