LECTURE 4. THE SUBCONVEXI1Y PROBLEM 255of Fourier coefficients of weight 0 Maass forms of level /! 1 /! 2 (taken at cusps not
necessarily equal to oo); then one can take advantage of the best known bounds for
the Fourier coefficients (which come from H 2 (B)) to improve the results. However,
as we shall see, there is a more direct way to make the connection between shifted
convolutions sums and Maass forms.4.4.1.1. Other Applications of the a-symbol Method. The delta-symbol method is
pretty elementary, and as such can be used for a variety of problems:- it was used by Duke/Iwaniec to provide non-trivial bounds for Fourier
coefficients of Maass forms (although weaker than the current bounds),
as well as to prove analytic continuation beyond the domain of absolute
convergence of certain L-series [Dul, Dul2, Dul3]. - Another interesting example is the following result of N. Pitt [Pi], which
solves the SCP for a GL 2 form with the simplest GL 3 Eisenstein series:
Theorem 4.9. Let g E sr(l) be a primitive holomorphic cusp form. Then
for X > 1 and 1 ~ I! < X^1124 , for any r:: > 0 one hasL Ag(n)T3(/!n - 1) «c:,g x71/72+c:.
l~n~X
The proof of this result is somewhat difficult and uses among other
things good bounds for sums of Kloosterman sums, following deep results
in the theory of exponential sums coming from the works of Deligne,
Bombieri, Adolphson-Sperber and Katz.- This method, combined with (the deeper) Theorem 4. 7, enables one to
evaluate asymptotically quite general shifted convolution type sums, i.e.
sums of the form
~~ m1 ni m 2 n 2
~~ Cl'.m1f3m2 V(M1' Ni' M2' N2);
m1n.1 -m2n2=h
here (CY.mi) , (f3m 2 ) denotes two arbitrary bounded sequences of complex
numbers; Vis a smooth function compactly supported on [-1, 1]^4 , where
the M i, N 1 close to each other (in the logarithmic scale), and (the crucial
point) where the smooth variables n 1 , n 2 can be taken a little smaller than
the non-smooth ones. Such sums lie at the heart of the proof of Theorem
4.1.
4.4.2. The Shifted Convolution Problem via Spectral Methods
In [Se2], Selberg proposed an approach to handle the SCP using the spectral de-
composition of a product of two modular forms; his suggestion was pursued suc-
cessfully in very specific cases in works of Good, Jutila, Motohashi, and others
[Go, TV, Ju2, Ju3, Mot, JM], but the first general treatment occurred in the work of
Sarnak, who applied this technique to solve the SCP for Rankin-Selberg £-functions
in the spectral aspect[Sa3, Sa4]:Theorem 4.10. [Sa4] Let g E Sk'(q', x', it') be a fixed cusp form and f E s r(q, X) a
primitive holomorphic form of weight k. Then