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: 3764332115

ISBN-13: 9783764332112

This ebook offers a large, straightforward advent 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. all the twelve chapters specializes in a selected subject dedicated to distinctive situations of this system. The booklet is acceptable for graduate scholars and researchers.

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**Additional resources for An introduction to the Langlands program**

**Example text**

Next, we prove a useful boundedness result for quotient sheaves of a given sheaf. 9 (Grothendieck) — Let P be a polynomial and ρ an integer. Then there is a constant C depending only on P and ρ such that the following holds: if X is a projective k-scheme with a very ample line bundle O(1), E is a d-dimensional sheaf with Hilbert polynomial P and Mumford-Castelnuovo regularity reg(E) ≤ ρ and if F is a purely d-dimensional quotient sheaf of E then μ ˆ(F ) ≥ C. Moreover, the family of purely d-dimensional quotients F with μ ˆ(F ) bounded from above is bounded.

G −1 = E/E1 . If Ei+1 ⊂ E denotes the pre-image of Gi , all that is left is to show that p(E1 ) > p(E2 /E1 ). But if this were false, we would have p(E2 ) ≥ p(E1 ) contradicting the maximality of E1 . For the uniqueness part assume that E• and E• are two Harder-Narasimhan filtrations. Without loss of generality p(E1 ) ≥ p(E1 ). Let j be minimal with E1 ⊂ Ej . Then the composition E1 → Ej → Ej /Ej−1 is a non-trivial homomorphism of semistable sheaves. 7. Hence, equality holds everywhere, implying j = 1 so that E1 ⊂ E1 .

In the second appendix to this chapter we prove a theorem due to Langton, which roughly says that the moduli functor of semistable sheaves is proper (cf. 2). 1 Flat Families and Determinants Let f : X → S be a morphism of finite type of Noetherian schemes. If g : T → S is an Sscheme we will use the notation XT for the fibre product T ×S X, and gX : XT → X and fT : XT → T for the natural projections. For s ∈ S the fibre f −1 (s) = Spec(k(s)) ×S X ∗ is denoted Xs . Similarly, if F is a coherent OX -module, we write FT := gX F and Fs = F |Xs .

### 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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