1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 4. THE SUBCONVEXITY PROBLEM 247 The second possibility of choosing a smaller subfamily of F containing n 0 , may be avai ...

248 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS which shows that for unramified p, >. 1 (p) and >. 1 (p ...

LECTURE 4. THE SUBCONVEXITY PROBLEM L i/2 «e: L lcil 2 + (qL)"'-( L lcil) 2 q l~L l~L ( l,q)=l (l,q)=l If we take now ( c1) to b ...

250 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS By elementary techniques (Cauchy/Schwarz, the Polya/Vinogrado ...

LECTURE 4. THE SUBCONVEXITY PROBLEM 251 of Dirichlet characters of modulus q and form the averaged mean square LIL:v(g, x)l^2 I ...

252 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF L -FUNCTIONS 4.4. The Shifted Convolution Problem Consider g a primitive ...

LECTURE 4. THE SUBCONVEXI1Y PROBLEM 253 limitation for many analytic applications. By exploiting the symmetry of the set of divi ...

254 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS It turns out that the best choice for U is U = min(X, Y) = X ...

LECTURE 4. THE SUBCONVEXI1Y PROBLEM 255 of Fourier coefficients of weight 0 Maass forms of level /! 1 /! 2 (taken at cusps not n ...

256 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS We present this approach for g E Sk ( q', x) holomorphic. It ...

LECTURE 4. THE SUBCONVEXITY PROBLEM 257 Finally by the unfolding method one has (u (. s ) u·)= 2s-1;oj(h)r(s-~+itj)r(s-~-itj) h ...

258 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS one obtains the averaged bound for T ~ 1: (4.21) L l(uj, V)l^ ...

LECTURE 4. THE SUBCONVEXITY PROBLEM 259 4.4.4. Integral of products of eigenfunctions As we have seen, the key point in the abov ...

260 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £ -FUNCTIONS holomorphic of weight k and level q', the bound (4.21) gives ...

LECTURE 4. THE SUBCONVEXITY PROBLEM 261 in the special case f, 1 R 2 = 1 and N = q'. By averaging over the primitive forms of le ...

262 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £ -FUNCTIONS work of Duke/ Friedlander/ Iwaniec [DFI3] for f a holomorphi ...

LECTURE 4. THE SUBCONVEXITY PROBLEM 263 (say) for La small power of Q, on average over F. The multiplicative properties of >. ...

264 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS Here x'(R2)Bxx'U\ m - fz n, O; c) = x(f 2)Gxx'(f 1f 2m - n ; ...

LECTURE 4. THE SUBCONVEXIlY PROBLEM 265 shows that the h =I= 0 terms of ( 4.31) contribute (essentially) « Qc:LL1~Llc1l 2 r:;;-q ...

266 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS bound of Theorem 4.8 replaced by Burgess's bound for Dirichle ...