An Introduction to the Langlands Program by Joseph Bernstein, Stephen Gelbart, S.S. Kudla, E. Kowalski,

Algebraic Geometry

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

For the prior numerous many years the speculation of automorphic types has turn into a big point of interest of improvement in quantity concept and algebraic geometry, with functions in lots of different components, together with combinatorics and mathematical physics.

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

Key positive factors of this self-contained presentation:

various parts in quantity thought from the classical zeta functionality as much as the Langlands software are coated.

The exposition is systematic, with each one bankruptcy targeting a selected subject dedicated to designated instances of this system:

• simple zeta functionality of Riemann and its generalizations to Dirichlet and Hecke L-functions, classification box idea 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) 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)) results in Langlands idea for GL(n) and the significance of the Langlands twin workforce (J.W. Cogdell)

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

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

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30 E. 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.

Since the inverse of the gamma factor is known to be an entire function of order = 1, it follows that L(x, s) is also. This is actually very far from the truth. Of course, for Re(s) > 1, where the series converges absolutely, it follows that L(x, s) « I, uniformly for Re(s) > 1 + 8, 8 > 0, for any x, the implied constant depending only on K. For Re(s) < 0, on the other hand, this implies, thanks to the functional equation, Stirling's formula and the 8Multiply by s(l - s) if X = I. 2. Elementary Theory of L-Functions II 33 shape of the gamma factor, that L(x,s) « lsl(l-a)/ 2 , uniformly fora =Re(s) < -8, 8 > 0.

Definition. : 1 an integer and x a Dirichlet character modulo q. :if = Af and f I y = x(a)f, for y E ro(q). 1. Taking y = -1, on gets the relation f = f lk y = x ( -1)( -1)k f, so there can exist nonzero holomorphic modular forms only if the character satisfies the consistency condition x (-1) = ( -1 )k, which is tacitly assumed to be the case in what follows. Similarly for automorphic functions, we must have x (-1) = 1. 3. Classical Automorphic Forms 47 One can also define nonholomorphic forms of weight k =F 0, using a modified differential operator.

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