A sieve method for shifted convolution sums

Abstract

We study the average size of shifted convolution summation terms related to the problem of Quantum Unique Ergodicity on SL2( )\À. Establishing an upper-bound sieve method for handling such sums, we achieve an unconditional result which suggests that the average size of the summation terms should be sufficient in application to Quantum Unique Ergodicity. In other words, cancellations among the summation terms, although welcomed, may not be required. Furthermore, the sieve method may be applied to shifted sums of other multiplicative functions with similar results under suitable conditions.

Shifted convolution sums for GL(3)×GL(2)

“…Our job now is to get a nontrivial estimate for (13), beyond square-root cancellation in the character sum S(m 1 , m 2 , n, h; q). For h = 0, the zero shift, the character sum S(m 1 , m 2 , n, 0; q) can be evaluated precisely, and then one can use the large sieve inequality of Duke, Friedlander and Iwaniec [8] for Kloosterman fractions to get extra cancellation on the sum over n and m. Alternatively one can use reciprocity and then Voronoi yet again on the sum over m 2 , to get a much better result.…”

“…Non-trivial bound of this sum often has deep implications, e.g. subconvexity and equidistribution (QUE) (see [2], [6], [7], [11], [12], [13], [14], [16], [17], [18], [20], [24]). In this paper we will consider a higher rank analogue -…”

Strong orthogonality between the Möbius function, additive characters and Fourier coefficients of cusp forms

“…λ F (n) n (k−1)/2 . By the above proposition, (16) and (17), for any primitive Maass cusp form f , whether or not it is induced from a holomorphic form, we have that…”

Mass equidistribution for automorphic forms of cohomological type on 𝐺𝐿₂

“…However, under certain natural assumptions about the distribution of λ f (p) it may be shown that M k (f ) is small; more precisely, in [14], Holowinsky shows that if neither…”