By Joseph Bernstein, Stephen Gelbart, S.S. Kudla, E. Kowalski, E. de Shalit, D. Gaitsgory, J.W. Cogdell, D. Bump
For the earlier numerous a long time the idea of automorphic varieties has turn into a tremendous point of interest of improvement in quantity thought and algebraic geometry, with functions in lots of varied components, together with combinatorics and mathematical physics.
The twelve chapters of this monograph current a wide, undemanding advent to the Langlands application, that's, the speculation of automorphic varieties and its reference to the idea of L-functions and different fields of arithmetic.
Key good points of this self-contained presentation:
various components in quantity concept from the classical zeta functionality as much as the Langlands application are coated.
The exposition is systematic, with each one bankruptcy concentrating on a specific subject dedicated to detailed instances of this system:
• easy zeta functionality of Riemann and its generalizations to Dirichlet and Hecke L-functions, type box concept and a few issues on classical automorphic functions (E. Kowalski)
• A examine 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 is the way to learn automorphic representations (D. Bump)
• dialogue of cuspidal automorphic representations of GL(2,(A)) ends up in Langlands thought for GL(n) and the significance of the Langlands twin staff (J.W. Cogdell)
• An advent to the geometric Langlands software, a brand new and lively region of analysis that allows utilizing robust tools of algebraic geometry to build automorphic sheaves (D. Gaitsgory)
Graduate scholars and researchers will take advantage of this gorgeous text.
Read Online or Download An Introduction to the Langlands Program PDF
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Extra info for An Introduction to the Langlands Program
2m L(x, s dw + w)G(w)-. 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.
One can show easily that the corresponding real character modulo m is unique. 3. The classical case is when K = Q and then the possible existence of f3t for a primitive odd quadratic character X modulo q is directly related to the possibility that the class number of the corresponding imaginary quadratic field K = Q(~) is very small. 5). Improving the "trivial bound" h(K) » q- 112 is extremely hard, although the Generalized Riemann Hypothesis implies that, in fact, L(x, 1) » - 1- , logq hence h(K) » ,;q .
This belongs historically to the early 20th century and the most important names for us are Eisenstein, Poincare, Heeke, Ramanujan, Petersson and (later) Maass and Selberg. We will speak simultaneously of holomorphic modular forms and of nonholomorphic (Maass) forms, although the latter were introduced quite a bit later: this will strongly motivate the development of more group-theoretic methods to unify the work being done. For simplicity we will mostly restrict our attention to congruence subgroups of SL(2, Z), which is arithmetically the most important.
An Introduction to the Langlands Program by Joseph Bernstein, Stephen Gelbart, S.S. Kudla, E. Kowalski, E. de Shalit, D. Gaitsgory, J.W. Cogdell, D. Bump