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

ISBN-10: 0817632115

ISBN-13: 9780817632113

ISBN-10: 0817682260

ISBN-13: 9780817682262

ISBN-10: 3764332115

ISBN-13: 9783764332112

For the earlier a number of many years the idea of automorphic types has turn into an important point of interest of improvement in quantity concept and algebraic geometry, with purposes in lots of diversified parts, together with combinatorics and mathematical physics.

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

Key positive aspects of this self-contained presentation:

a number of parts in quantity thought from the classical zeta functionality as much as the Langlands software are lined.

The exposition is systematic, with every one bankruptcy concentrating on a selected subject dedicated to certain circumstances of this system:

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

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

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

• Selberg's thought of the hint formulation, that is how to examine automorphic representations (D. Bump)

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

• An creation to the geometric Langlands software, a brand new and lively region of analysis that enables utilizing strong tools of algebraic geometry to build automorphic sheaves (D. Gaitsgory)

Graduate scholars and researchers will reap the benefits of this pretty text.

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Additional resources for An Introduction to the Langlands Program

Example text

Both holomorphic and Maass forms can be most convincingly put into a single framework through the study of the representation theory of G L(2, R) (or of the adele group G L(2, A) in the arithmetic case). Using the definition above, one can impose more regularity conditions at the cusps. Definition. , an automorphic function). -periodic function fa = f lk a a is of moderate growth at infinity. , Maass forms). , sJ. , Maass cusp forms). 2. Other equivalent formulations can be given. ,).. =F 0) and in L 2 (fo(q)\H) (with respect to the hyperbolic measure).

Then A (s; a) admits analytic continuation to a meromorphic function on C with simple poles at s = 1 and at s = 0 and it satisfies the functional equation · A(s; a)= IDI 112-s A(l- s; (aD)- 1), where D is the ideal class of the different of K /Q. JIDT . Summing over a E H(K), this proposition implies the analytic continuation and functional equation of {K (s) as stated in Section 4. 2), are absolutely convergent for Re(s) > 1, and since all partial zeta functions have the same residue at s = 1.

2. , [Wa]). C. 4] prove that the denominator of -bk/ k contains all the primes p such that p - 1 I 2k, and only them, so that for a p in the denominator we have xk- 1 = x- 1 (mod p). Hence even the "poles" modulo primes of ~(1 - k) are still "explained" by the divergence of the harmonic series! In the other direction, maybe it is not so surprising that the numerator of Bernoulli numbers should remain so mysterious. We finally only mention that the congruence properties of the values of the zeta function at negative integers were the motivation for the discovery by Leopoldt of the p-adic zeta function, later much generalized by others to p-adic L-functions of various kinds.

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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

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