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 speculation of automorphic types has turn into an important point of interest of improvement in quantity concept and algebraic geometry, with purposes in lots of different components, together with combinatorics and mathematical physics.

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

*Key positive factors of this self-contained presentation:*

numerous components in quantity idea from the classical zeta functionality as much as the Langlands application are coated.

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

• simple zeta functionality of Riemann and its generalizations to Dirichlet and Hecke L-functions, type box conception 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) capabilities, bringing into play the speculation of representations **(S.S. Kudla)**

• Selberg's idea of the hint formulation, that's the way to examine 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 application, a brand new and energetic sector of study that enables 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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**Additional resources for An Introduction to the Langlands Program**

**Example text**

1. Let K be a number field of degree d = [K : Q], X a primitive Heeke character of K of modulus m. 2) 2, the estimate being uniform in all parameters. 3) IPII+t: is absolutely convergent. The proof of this asymptotic formula is straightforward in principle: one applies Cauchy's Theorem to the logarithmic derivative of the function A(x, s) (which has the same zeros as L(x, s)) in the rectangle [ -1/2, 3/2] x [- T, T]. The major contribution comes from the gamma factor for X. 3) is actually also part of this theory).

L. acting on L 2 (f\H) satisfies AJ ;;:;: 1I 4. 1 ;;:;: 3I 16. This is an arithmetic statement because it is easy to construct noncongruence subgroups for which there are arbitrarily many exceptional eigenvalues. 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).

3. Classical Automorphic Forms 47 One can also define nonholomorphic forms of weight k =F 0, using a modified differential operator. 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.