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 idea of automorphic kinds has develop into a tremendous point of interest of improvement in quantity idea and algebraic geometry, with functions in lots of assorted parts, together with combinatorics and mathematical physics.

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

*Key good points of this self-contained presentation:*

quite a few components in quantity thought 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 exact instances of this system:

• easy zeta functionality of Riemann and its generalizations to Dirichlet and Hecke L-functions, category box conception 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 idea of the hint formulation, that is how to research 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 creation to the geometric Langlands software, a brand new and lively sector of study that allows utilizing robust equipment of algebraic geometry to build automorphic sheaves **(D. Gaitsgory)**

Graduate scholars and researchers will make the most of this pretty text.

**Read Online or Download An Introduction to the Langlands Program PDF**

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**Extra resources for An Introduction to the Langlands Program**

**Example text**

Z; AWJ, AW2) = ).. -k f(z; WI, W2), and thus f(u; WJ, w2) = w:;_kg(ufw2, WJ/W2), for some function g defined on C x (H U H), which restricted to C x H is such that g(z; yr) = (cr + dlg(z; r) for y E SL(2, Z). 1. The Weierstrass p-function of the lattice A is s:> (z; WI • W2) = 2z1 + L we A w~O ( 1 )2 (z-w 2w1 ) ' 3. Classical Automorphic Forms 43 amd it is A -invariant (this is not so obvious: differentiate to see that p' is definitely A -invariant, then integrate). Clearly, p(Az; AWI, A2Wz) =A - 2p(z; w1, a>z).

All this is largely based on the interpretation of Xo(q) as a moduli space: it "classifies" pairs (E, H), where E is an elliptic curve and H is a cyclic subgroup of order q on E. The image in Xo(q) of the point z E H then corresponds to the pair (C/(Z 61 zZ), <1/q> ). These are very important aspects that will be discussed in further chapters, and it is the key to many of the deep arithmetical properties of modular forms. This is related to our third motivating example for modular forms. 48 E. Kowalski An easy example of those links is the important isomorphism Sz(q) ~ r(Xo(q), QI) If ~ f(z)dz between the space of weight 2 cusp forms of level q (with trivial nebentypus) and the space of holomorphic 1-forms on the modular curve Xo(q) (not Yo(q)).

The last paragraph of this remark shows (since Xo(q) is compact so the space of global differentials r(Xo(q), QJ) is finite dimensional) that dime Sz(q) < +oo. (q,x)) is finite. 6]: for instance dim Sz(q) is the genus of Xo(q), by the above, which can be computedexplicitly,usingthenatural(ramified)coveringXo(q) ~ Xo(l) = P(l) induced by the inclusion ro(q) < ro(l). With small fluctuations, the dimension increases as the level and the weight do: dim Mk(q) is of size (roughly) qkll2, so there are many modular forms as soon as the weight or the level is not too small (for instance, the genus of Xo(q) is> 1 when q > 49, or if q is prime > 19).