An Introduction to the Langlands Program by Joseph Bernstein, Stephen Gelbart, S.S. Kudla, E. Kowalski,

By Joseph Bernstein, Stephen Gelbart, S.S. Kudla, E. Kowalski, E. de Shalit, D. Gaitsgory, J.W. Cogdell, D. Bump

This e-book offers a vast, simple advent to the Langlands software, that's, the speculation of automorphic varieties and its reference to the idea of L-functions and different fields of arithmetic. all of the twelve chapters specializes in a specific subject dedicated to unique situations of this system. The e-book is appropriate for graduate scholars and researchers.

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6. We now come to theta functions, with notation as in [11, Chapter 10], in a fairly general situation. Let A be a symmetric positive definite (r x r)matrix with integral coefficients, with all diagonal coefficients even, and N ~ 1 any integer such that N A -I has integer coefficients. We let A[x] = t xAx, for x ERr be the associated quadratic form. Assume the number of variables r is even. 6 Let 8(z; A) =L e(A[m]z/2). 7. We have 8(z; a) E Mr;2(2N). 5). For instance, let A4 = diag(2, 2, 2, 2). 1).

21rr f (-3) dw w L(x, s + w)G(w)-. 36 E. Kowalski Now apply the functional equation of L(x, s): the last integral (say J) becomes f 1 = s(x)(IDINm) 112-s 1 . 2irr f(~oo. 1- s- w) f(~oo. s + w) L(x. - 2m f (3) f(~oo. 1- s- w) f(~00 , s + w) L(x. 1 - s + w)G(w) ( X IDINm )-w dw -. J (Na)l-s y Cl Hence the result since G(O) = 1. , the critical line is translated to Re(s) = 1/2) the points = 1 is the only possible pole for an (automorphic) £-function, and further that such a pole is always accounted for by the simple pole of the Riemann zeta function, in the sense that the L-function L(f, s) has a factorization where L Cf1, s) is another L- function which is entire.

2, exceptional eigenvalues for fo(q) have an effect such as the Landau-Siegel zero for Dirichlet characters: the uniformity in q (say in counting solutions to ad- be = h with c = 0 (mod q) with a 2 + b 2 + c 2 + d 2 ~ X) is affected by the presence of "many" exceptional eigenvalues (the closer to 0, the worse the effect). Hence Selberg's theorem indicates that the situation is a little bit better controlled. 2 below), and indeed the significant improvements to the 3116 bound proved by Luo, Rudnick and Sarnak [LRS] is based on this analogy.

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