A large sieve inequality of Elliott–Montgomery–Vaughan type for automorphic forms on GL$_3$

Abstract

We get a large sieve inequality of Elliott-Montgomery-Vaughan type for automorphic forms on GL3 in short intervals. As an application, we give a statistical result of sign changes for Hecke eigenvalues Aj(p, p).

On Signs of Fourier Coefficients of Hecke-Maass Cusp Forms on $\mathrm{GL}_3$

“…We have the following result due to Xiao ja Xu [44,Theorem 1]. Let h T be the same weight function as in (1).…”

“…For the group SL 3 (Z), Steiger [43] considered signs of the real part of A(p, 1) and proved (with his vertical Sato-Tate law) that there is a positive density of Hecke-Maass cusp forms φ such that the real part of 1 A φ (p, 1) is positive for all primes p in some finite set. After that Xiao and Xu [44] considered the number of Hecke-Maass cusp forms φ such that A φ (p, p) has a given sign for any prime p lying in certain region. The latter result has since been generalised by Lau, Ng, Royer, and Wang [20] for the group SL n (Z) with n ≥ 4.…”

A large sieve inequality of Elliott–Montgomery–Vaughan type for Maass forms on GL($n,\mathbb R$) with applications

“…In the case of GL(2, R), there are fruitful results (for example, see [5], [12], [13]). In the case of GL(3, R), Steiger [14] proved that there is a positive proportion of Hecke-Maass forms φ with positive real part of A φ (p, 1) for a fixed prime p and Xiao and Xu [16] gave a statistical result on the signs of A φ (p κ1 , p κ2 ) + A φ (p κ2 , p κ1 ). Applying Theorem 1.1, we obtain the following result.…”

“…There are two main difficulties: the first one is that for n ≥ 3 the Hecke relations for GL(n, R) are much more complicated than those of GL(2, R), and the trace formula for GL(n, R) with n ≥ 3 is not as simple as the trace formula (say Kuznetsov's and Petersson's trace formulas) on GL(2, R). Recently, Xiao and Xu [16], using Kuznetsov's trace formula and Hecke's relations, made a breakthrough and obtained a large sieve inequality of EMV type to Maass forms on GL(3, R). Moreover, they also applied their large sieve inequality to get a statistical result of sign changes on the Hecke eigenvalues for GL (3, R).…”

“…Our main tool is the automorphic Plancherel density theorem -a recent great progress due to Matz and Templier [10]. We remark the use of properties of (degenerated) Schur's polynomials instead of Hecke's relations to avoid the complicated calculations as in [16]. More precisely the (degenerated) Schur polynomial is employed to evaluate the main term when applying the truncated trace formula [6,Corollary 3.3] since the main term in [6,Corollary 3.3] is expressed in the form of orbital integral involving the (degenerated) Schur polynomial by the work of Matz and Templier [10].…”