1549380232-Automorphic_Forms_and_Applications__Sarnak_

252 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF L -FUNCTIONS

4.4. The Shifted Convolution Problem

Consider g a primitive form, h =!= 0, £ 1 ,£ 2 two coprime integers and W(x,y ) a

bounded, smooth, compactly supported function in [X, 2 X] x [Y, 2 Y] for some

X, Y ~ 1 /2; to fix ideas we suppose also that W has well controlled derivatives, i.e.

(4.7)

One considers the following shifted convolution sums:

(4.8) L >- 9 (m)>- 9 (n)W(£1m,£2n) ,

which, using (2.17), are easily bounded by

"L.w(g,£1, £2, h) «c (Q 9 XY)E Max(X/£1, Y/£2).

The Shifted Convolution Problem (SCP) is to improve the trivial bound for rel-

evant choices of M, N, £; what makes it possible is the non-vanishing of h, so

that >- 9 (m) do not conspire with ,\ 9 (n) = ,\ 9 (e'7,- h). This problem has a long

history that which goes back to Ingham with the case of the Eisenstein series

g(z) = E'(z, 1 /2) and ,\ 9 (n) = T(n) the divisor function (this instance of the SCP is

the classical Shifted Divisor Problem). The strategy to solve the SCP is to "smooth

out" the condition £ 1 m - £ 2 n = h in L.w in order to "separate" the variable m from

n and to estimate the resulting sums. In this section we describe two techniques for

achieving this goal. These methods will provide a non-trivial bound uniformly for

Max(X/£ 1 , Y/£ 2 ) larger than some fixed (relatively large) powers of Q 9 and £ 1 £2,

respectively. This will be sufficient for the present subconvexity cases. However,

the question of the uniformity of these bounds with respect to the parameters of g

is very interesting and will be discussed later on.

Although the Shifted Convolution Problem looks rather technical, it is ubiqui-

tous in the resolution of the Subconvexity Problem for G £ 1 and G £ 2 £-functions.

Indeed, all the cases of subconvexity that are presented in the Subconvexity theo-

rem at the beginning of this lecture can be reduced to the resolution of the SCP for

some appropriate modular form g: for instance the identity

L(x, s )^2 = L x(n):(n) = L (x x E'(z, 1 /2), s)

n;,:l n

shows that the Subconvexity Problem for Dirichlet £ -function could follow from

a variant of the method presented in the previous section and the solution of an

instance of the SCP.

4.4.1. The J-symbol Method

The J -symbol method was developed in [DFil, DFI3] as a variant of the circle

method. Its main purpose is to express 6 ( n), the Dirac symbol at 0 (restricted to

the integers n in some given range: lnl ~ N ), in terms of additive characters. Of

course one could use the orthogonality relation

1~ nk

6n:=O(q) = -L., e( - )

q k(q) q

which provides an adequate expression as long as 2N ~ q; however, the conductor
q of the additive characters is large compared with N and this constitute a severe