1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 1 Mostly SL(2) 1.1. In this lecture we will describe, for the simplest group G = SL(2, Q), the phenomena of spectral ana ...

48 L. CLOZEL, SPECTRAL THEORY OF AUTOMORPHIC FORMS Note that E(z, ~+it) is not an £^2 -function on r\H; neither should it be sin ...

LECTURE 1. MOSTLY SL(2) 49 In all cases we see that p-^1 /^2 < lapl, l,8pl < p^112 ("Hecke's trivial estimate"). Now we ca ...

50 L. CLOZEL, SPECTRAL THEORY OF AUTOMORPHIC FORMS The theorem was proved by Eichler and Shumura for k = 2, then by Deligne for ...

LECTURE 1. MOSTLY SL(2) 51 This implies that '{Jp, tr.pp and Tp commute as endomorphisms of H^1. Moreover 'Pp verifies the equat ...

52 L. CLOZEL, SPECTRAL THEORY OF AUTOMORPHIC FORMS by GL(2, Zp)· To it is associated a Hecke matrix, that is, a diagonal matrix ...

LECTURE 1. MOSTLY SL(2) 53 There is an obvious notion of upper density, and Q(S) ;::: 1 - J(P - S); in fact Q(S) + J(P - S) = 1; ...

54 L. CLOZEL, SPECTRAL THEORY OF AUTOMORPHIC FORMS Lemma. - Assume G is a finite group and r an irreducible representation of G; ...

LECTURE 1. MOSTLY SL(2) 55 (1) Spectral information about differential operators on r\G(IR) - e.g. the Laplacian on r \G(IR)/ K= ...

56 L. CLOZEL, SPECTRAL THEORY OF AUTOMORPHIC FORMS To conclude, we refer the reader to the Corvallis volume [CJ, in particular t ...

LECTURE 2 Arthur's conjectures Lecture 2. The spectral decomposition of L^2 (G(Q)\G(A)): Arthur's conjectures 2.1. In this lectu ...

58 L. CLOZEL, SPECTRAL THEORY OF AUTOMORPHIC FORMS where B is a Borel subgroup, x is a quasi-character of T(Qp) ~ (Q; )", TC B i ...

LECTURE 2. THE SPECTRAL DECOMPOSITION OF L^2 (G(Q)\G(A)) 59 subspace of H^1 (Xo(N), Qe)).^2 Thus t 1 , t 2 are p-^1 /^2 x (Weil ...

60 L. CLOZEL, SPECTRAL THEORY OF AUTOMORPHIC FORMS SL(2, Z), n = @n v is the associated representation of GL(2, A), the correspo ...

LECTURE 2. THE SPECTRAL DECOMPOSITION OF L^2 (G(IQ)\ G(A)) 61 "purity + Lefschetz" behaviour onrepresentations of G(A) which hav ...

62 L. CLOZEL, SPECTRAL THEORY OF AUTOMORPHIC FORMS WIR being still the Weil group, with bounded restriction to WJR. In this case ...

LECTURE 2. THE SPECTRAL DECOMPOSITION OF L^2 (G(IQl)\G(A)) 63 The existence of (2.8) also has consequences if we impose conditio ...

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LECTURE 3 method Lecture 3. Known bounds for the cuspidal spectrum and the Burger-Sarnak Burger-Sarnak method 3.1. We first retu ...

66 L. CLOZEL, SPECTRAL THEORY OF AUTOMORPHIC FORMS conjecture (nv tempered) for primes where 1fv is ramified. We say no more, re ...