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

For the prior numerous a long time the speculation of automorphic kinds has develop into a big point of interest of improvement in quantity conception and algebraic geometry, with purposes in lots of diversified parts, together with combinatorics and mathematical physics.

The twelve chapters of this monograph current a vast, basic advent to the Langlands application, that's, the idea of automorphic varieties and its reference to the speculation of L-functions and different fields of arithmetic.

Key beneficial properties of this self-contained presentation:

numerous components in quantity concept from the classical zeta functionality as much as the Langlands software are lined.

The exposition is systematic, with every one bankruptcy concentrating on a specific subject dedicated to distinctive situations of this system:

• simple zeta functionality of Riemann and its generalizations to Dirichlet and Hecke L-functions, classification box thought and a few subject matters on classical automorphic functions (E. Kowalski)

• A learn of the conjectures of Artin and Shimura–Taniyama–Weil (E. de Shalit)

• An exam of classical modular (automorphic) L-functions as GL(2) services, bringing into play the speculation of representations (S.S. Kudla)

• Selberg's concept of the hint formulation, that's how to learn automorphic representations (D. Bump)

• dialogue of cuspidal automorphic representations of GL(2,(A)) ends up in Langlands concept for GL(n) and the significance of the Langlands twin workforce (J.W. Cogdell)

• An advent to the geometric Langlands software, a brand new and lively zone of study that allows utilizing robust tools of algebraic geometry to build automorphic sheaves (D. Gaitsgory)

Graduate scholars and researchers will make the most of this pretty text.

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Extra resources for An Introduction to the Langlands Program

Example text

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. lt is known also for Artin L-functions, using Brauer's Theorem and the non vanishing of Heeke L-functions at s = 1.

1. Let K be a number field, weight ~00 • Then we have L(x, s) « x a Heeke character to modulus m with (IDINm(1 +lsi))£ for any s > O,for 1/2 ~ Re(s) < 1, with an implied constant depending only on s. If X = 1, multiply on the left by (s - 1). This is called the Lindeli::if Conjecture. 2]). 2. Let K and x be as above and lets = a + it with 0 < a < 1. Let G(w) be any function real-valued on Rand holomorphic in the strip -4 < Re(s) < 4, with rapid decay as I lm(w)l ~ +oo, and such that l G(-w) = G(w), G(O) = 1, G(w)f(~ 00 , w) is holomorphicfor w = 0, -1, -2, -3.

S)- r(~oo. 1- s) ) L' - z;