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

This publication offers a large, easy creation to the Langlands application, that's, the speculation of automorphic kinds and its reference to the speculation of L-functions and different fields of arithmetic. all the twelve chapters makes a speciality of a selected subject dedicated to specific circumstances of this system. The ebook is acceptable for graduate scholars and researchers.

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Kowalski The best knpwn results-proved using the spectral theory of automorphic forms (see Chapter 3)-for primes in arithmetic progressions go beyond what is immediately provable from GRH (Bombieri-Friedlander-Iwaniec); they are commonly used in applications. The same results and difficulties occur in the Chebotarev density theorem for Artin £-functions, often exacerbated because the degree of interesting families of fields is larger than that of cyclotomic fields, making even the form of the prime ideal theorem based on GRH insufficient for applicat10ns.

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. The spectral analysis of the general Poincare series is the essence of the Kuznetsov-Bruggeman formula which is of great importance in the applications of Maass forms to analytic number theory (see [12, Chapter 9], [CP]). Properly speaking, we have defined the Poincare and Eisenstein series relative to the cusp oo.

Note that G may depend on s; however, very often the points is fixed in the applications and it is the character which varies). The point of this formula is that Vs and Ws can be easily estimated using contour shifts and the Stirling formula. , they decay very rapidly for y large. 1), one sees easily that the first sum is essentially a sum over ideals of norm N a ~ X, while the second is over ideals of norm Na ~ Y(l + lsl)dfZ (this means the tails are very small). 1) gives an expression of the L-function inside the critical strip by very rapidly convergent series.