1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 1. ANALYTIC PROPERTIES OF INDMDUAL £-FUNCTIONS 207 1.3.3.1. What do we mean by "nice family"? In practice one may procee ...

208 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £ -FUNCTIONS abuse of notation, we still denote by X· We take the x to be ...

LECTURE 1. ANALYTIC PROPERTIES OF INDMDUAL £-FUNCTIONS 209 one finds that = L q ; 1 L >-1r:~ ( n) Vi (;) - L L >-1r:~ ( n) ...

210 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS and hence ~ q - 1 {3 d^2 +1 {3 T1 + T2 = L - 2 O(QY^1 - +c ...

Lecture 2. A review of classical automorphic forms In this section we review the theory of GL 2 ,q-automorphic forms from the cl ...

212 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS for every 1 = ( : ~ ) E r, and that vanish at every cusp). Th ...

LECTURE 2. A REVIEW OF CLASSICAL AUTOMORPHIC FORMS 213 where r a denotes the stabilizer of the cusp a, and that a cusp a is sing ...

214 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS defines an isomorphism (which is in fact an isometry up to so ...

LECTURE 2. A REVIEW OF CLASSICAL AUTOMORPHIC FORMS 215 where o(it k) _ r(1/2 +it - k/2) ' - r(1/2 +it+ k/2)' then Qit,k is null ...

216 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS S'k(q, x, it)) is determined up to scalars by all but finitel ...

LECTURE 2. A REVIEW OF CLASSICAL AUTOMORPHIC FORMS 217 In particular, under H 2 (B) one has 1 1 (2.11) AJ = 4 + t^2 ~ 4 -e^2 , a ...

218 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF L-FUNCTIONS type theorem for L( 7r f ®ir f, s) (which is known, see Secti ...

LECTURE 2. A REVIEW OF CLASSICAL AUTOMORPHIC FORMS 219 Mellin transform, the J Bessel function appearing as the inverse Mellin t ...

220 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £ -FUNCTIONS Theorem 2.1. Fork? 2, let Bk(q, x ) denotes an orthogonal ba ...

LECTURE 2. A REVIEW OF CLASSICAL AUTOMORPHIC FORMS 221 In fact, this formula is not quite sufficient for all purposes. In order ...

222 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS or the bounds for I given in (2.30), one can derive the follo ...

Lecture 3. Large sieve inequalities 3.1. The large sieve In this lecture we describe in greater detail the concept of quasi-orth ...

224 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS Remark 3.1. There is a natural limitation on the possible siz ...

LECTURE 3. LARGE SIEVE INEQUALITIES 225 3.1.2. Large sieve inequalities for characters Simple examples of large sieve type inequ ...

226 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS Remark 3.2. By using the spectral analysis on modular forms ( ...