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

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

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

Chapter I specializes in 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 turn out the analytic continuation and useful 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 ebook may be of worth to undergraduate and graduate arithmetic scholars.

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In general, the full Dynkin diagram corresponds to the group Η = S X n + i , and the diagram determined by the labelled simple roots corresponds to the Levi component (GLn) of the parabolic subgroup Ρ of Η. The representation r\ = r is the standard representation pn : GLn{ (C ) — • GLn( (C ) with highest weight λχ, equal to the fundamental weight 6\. A slightly different example is (iii) on page 47 of [Lai]: 0 0 Oil ... 0—0—0 ocm-i a m + ... i r = 1 Here Η = G L n + i , and r : GLm(

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. ) We first describe the L-group by making explicit the action of on the simple roots and, more generally, on φο(ϋ). ι(α, 6, α ) = (a\ + ia2)(bi - 1 α2(α, δ, α ) = (δι + ib2)(ai — ^2)5 a n < l — ia>2)- Then we compute 1 (σαι)(α, 6, α " ) = (αχ - i ^ X ^ l + ih) l = a2{a,b,ä~ ) , and similarly, σα2 = αχ. (g) = t w g~^w described above clearly induces this same automorphism of L ^ o ( ^ G ° ) .

Fix an unramified quadratic extension Κ of the local field F , and V a unitary 3-dimensional space over K. We pick a basis for V so thee matrix 0 0 1 corresponding to the hermitian form on V is J = I 0 1 0 , and let LI G = ,0 in GLs(K) belongs to G if and only if (gx,gy) = (x,y) for all x,g in 3 (Here ( , ) denotes the Hermitian form defined on AT χ i f t xJy. , g 3 3 K. ) To see that G is a (quasi-split) group defined over j P , it is convenient to pick a basis { l , i } for Κ over F such that i non-trivial automorphism σ of Gal(K/F) If g in GL$(K) 2 £ J P , and such that the takes χ + iy in Κ to χ — iy.

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