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

Abstract

For the shifted convolution sumwhere λ 1 (1, m) are the Fourier coefficients of a SL(3, Z) Maass form π 1 , and λ 2 (m) are those of a SL(2, Z) Maass or holomorphic form π 2 , and 1 ≤ |h| ≪ X 1+ε , we establish the bound D h (X) ≪π 1 ,π 2 ,ε X 1− 1 20 +ε . The bound is uniform with respect to the shift h.1991 Mathematics Subject Classification. 11F66, 11M41.

Trace functions over finite fields and their applications

“…Thus Theorem 5.4 follows also from the fact that the group G FT ψ (K 2 ) is trivial, which is not very difficult to prove. Interestingly, other correlations sums attached to the same sheaf appeared in other papers in the literature: we are aware of its occurrence in works of Pitt [25] and Munshi [24].…”

A study in sums of products

“…[1,Corollary 11.24,Lemma 12.8]); -cases where k = 2 and γ 1 , γ 2 are diagonal are found in the thesis of Michel and his subsequent papers (e.g. [2]); -for k = 2, γ 1 = 1 and h = 0, we obtain the general 'correlation sums' (for the Fourier transform of K) defined in [3]; these are crucial to our works [3][4][5]; -special cases of this situation of correlation sums can be found (sometimes implicitly) in earlier works of Iwaniec [6], Pitt [7] and Munshi [8]; -the case k = 2, γ 1 and γ 2 diagonal, h arbitrary and K a Kloosterman sum in two variables (or a variant with K a Kloosterman sum in one variable and γ 1 , γ 2 not upper triangular) occurs in the work of Friedlander & Iwaniec [9], and it is also used in the work of Zhang [10] on gaps between primes; -cases where k is arbitrary, the γ i are upper triangular and distinct, and h may be non-zero appear in the work of Fouvry et al [11,Lemma 2.1], indeed in a form involving different trace functions K i (γ i • x) related to symmetric powers of Kloosterman sums; -the sums for k arbitrary and h = 0, with K a hyper-Kloosterman sum appear in the works of Fouvry et al [12] and Kowalski & Ricotta [13] (with γ i diagonal); -this last case, but with arbitrary h and the γ i being translations also appears in the work of Irving [14], and (for very different reasons) in work of Kowalski & Sawin [15]; and -another instance, with k = 4, h arbitrary and γ i upper triangular, occurs in the work of Blomer & Milićević [16,§11].…”

Shifted convolution sums involving theta series

“…The shifted convolution sum n≤X a 1 (n + h)a 2 (n) where a 1 (n) and a 2 (n) are two arithmetic functions and h ≥ 1 an integer, is an interesting and important object in analytic number theory and has been studied intensively by many authors with various applications (see for example [2], [6], [11], [13], [15]). For the arithmetic function r ℓ (n) = # (n 1 , n 2 , .…”

“…To prove Theorem 1, we apply Jutila's variation of the circle method (see [9]) which gives an approximation for where L = q∈Q φ(q). Then I Q,δ (x) is an approximation for I [0,1] (x) in the following sense (see Lemma 4 in Munshi [15]):…”