1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 3. LOCAL £-FUNCTIONS 127 Corollary. The functions e(s; W, W') and e(s; W, W', <f?) are entire functions, bounded in v ...

128 J.W. COGDELL, £-FUNCTIONS FOR GLn Let us say a few words about the proof of this proposition, because it is another applicat ...

LECTURE 3. LOCAL £-FUNCTIONS 129 or w(l -s; w, W', ~) - '(-l)n-1 ( I ·'·) w(s; w, W', ) L ( l - s,7r x 7r^1 ) - w € s, 7r x 7r ' ...

130 J .W. COGDELL, £-FUNCTIONS FOR GLn of unramified representations associates to each of these representation the semi- simple ...

LECTURE 3. LOCAL £-FUNCTIONS where sr(A) is the rth_symmetric power of the matrix A, and r=O for any matrix A. Then we quickly a ...

132 J.W. COGDELL, £-FUNCTIONS FOR GLn and can be found in [15,23]. In essence, in the family of integrals w(s; W, W', ), if (O) ...

LECTURE 3. LOCAL L-FUNCTIONS 133 More generally, if in addition 1f^1 is an irreducible generic representation of GLm(k) then for ...

134 J.W. COGDELL, £-FUNCTIONS FOR GLn Proposition 3.6. Let n be a generic representation of GLn(k). Then there is a finite set o ...

LECTURE 3. LOCAL £-FUNCTIONS 135 Towards this end, Jacquet and Shalika enlarge the allowable space of local inte- grals. Let A a ...

136 J.W. COGDELL, £-FUNCTIONS FOR GLn As a final result, let us note that in [16] it is established that the linear function- al ...

Lecture 4. Global £-functions Once again, we let k be a global field, A. its ring of adeles, and fix a non-trivial continuous ad ...

138 J.W. COGDELL, £-FUNCTIONS FOR GLn Theorem 4.1. The global £ - functions L(s, -rr x -rr') are nice in the sense that (1) L(s, ...

LECTURE 4. GLOBAL £-FUNCTIONS 139 However, by the local functional equations, for each v ES we have _ (l _. ( )W W') _ ~(1 -s; p ...

140 J.W. COGDELL, £-FUNCTIONS FOR GLn seems to be no obstruction to carrying this out, and then we obtain boundedness in vertica ...

LECTURE 4. GLOBAL £-FUNCTIONS 141 Observation. For n and n' cuspidal representations of GLn(A), L(s, n x n') has a pole at s = 1 ...

142 J.W. COGDELL, £-FUNCTIONS FOR GLn work with the incomplete £-function Ls(s,1f x 7r^1 ) where Sis as at the beginning of this ...

LECTURE 4. GLOBAL £-FUNCTIONS 143 Conjecture (GRC). Let 7r be a (unitary) cuspidal representation of GLn(A) with decomposition 7 ...

144 J.W. COGDELL, £-FUNCTIONS FOR GLn A simple generalization of the method used in [56] gives the analogous estimate at the arc ...

LECTURE 4. GLOBAL £-FUNCTIONS 145 Even in the simplest case of n = 1 and 7r = 1 the trivial representation this reduces to the R ...

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