By Joseph Bernstein, Stephen Gelbart, S.S. Kudla, E. Kowalski, E. de Shalit, D. Gaitsgory, J.W. Cogdell, D. Bump
For the previous numerous a long time the idea of automorphic varieties has turn into a massive point of interest of improvement in quantity conception and algebraic geometry, with purposes in lots of varied parts, together with combinatorics and mathematical physics.
The twelve chapters of this monograph current a vast, basic advent to the Langlands software, that's, the speculation of automorphic kinds and its reference to the speculation of L-functions and different fields of arithmetic.
Key positive aspects of this self-contained presentation:
quite a few components in quantity conception from the classical zeta functionality as much as the Langlands software are lined.
The exposition is systematic, with each one bankruptcy concentrating on a selected subject dedicated to targeted instances of this system:
• simple zeta functionality of Riemann and its generalizations to Dirichlet and Hecke L-functions, category box thought and a few themes 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) capabilities, bringing into play the idea of representations (S.S. Kudla)
• Selberg's thought 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 crew (J.W. Cogdell)
• An creation to the geometric Langlands software, a brand new and energetic quarter of study that enables utilizing robust equipment of algebraic geometry to build automorphic sheaves (D. Gaitsgory)
Graduate scholars and researchers will make the most of this gorgeous text.
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Extra resources for An Introduction to the Langlands Program
In the case of q = 1, we get the classical Eisenstein series Ek(Z) = L (a,b)=J 1 (az + b)k. 3) yEf oo \fo(q) 5The problem of bounding from above in a precise way the multiplicity of Maass cusp forms is one of the most inscrutable open problems in analytic number theory. 50 E. Kowalski (noticethatim(yz) = lcz+dl- 2 fory E SL(2, R)). l. = s(l- s) and it is quite easy to check that it has polynomial growth at the cusps. 2 for a very short introduction). In general, it is often convenient to write any eigenvalue in this way (so two values of s exist for a given )..
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). 2]). 2) nEZ and f being holomorphic means an = 0 for n < 0, since le(nz)l = exp( -2rrny) is not polynomially bounded for n < 0. 3. Let Yo (q) = r o(q) \H.
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. 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).
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