Selberg orthogonality for half-integral weight modular forms

TL;DR

This paper investigates Selberg orthogonality for half-integral weight modular forms, demonstrating decorrelation results for Fourier coefficients, automorphic periods, and Tate--Shafarevich groups under GRH and Ramanujan conjecture assumptions.

Contribution

It establishes a variant of Selberg orthogonality for half-integral weight forms and applies it to several arithmetic problems involving automorphic forms and elliptic curves.

Findings

Decorrelation of Fourier coefficients of half-integral weight modular forms.

Decorrelation of automorphic periods over prime quadratic fields.

Decorrelation of Tate--Shafarevich group orders under prime quadratic twists.

Abstract

The Keating--Snaith conjecture for orthogonal families may be viewed as analogous to a Gaussian distribution with a negative mean, and the possibility that mixed moments resemble a composition of independent moments, these two insights were combined and applied in Lester and Radziwi{\l\l}'s proof of quantum unique ergodicity for half-integral weight automorphic forms, via Soundararajan's method under the Generalized Riemann Hypothesis (GRH). This observation also yields a crucial and nontrivial saving in the resolution of certain arithmetic problems. Inspired by this, we select a series of typical mixed orthogonal families of $L$-functions: $\mathrm{GL}_2$ quadratic twisted families, Gao and Zhao established a sharp upper bound by building upon Harper's method, and one can replace square-free numbers with primes in this argument. Under the assumptions of the GRH and the Generalized Ramanujan Conjecture, we present the following three arithmetic applications: i) The decorrelation of Fourier coefficients of half-integral weight modular forms, specifically, a variant of Selberg orthogonality for distinct half-integral weight modular forms. ii) The decorrelation of automorphic periods averaged over prime imaginary quadratic fields. iii) The decorrelation of the analytic orders of isotropy subgroups of Tate--Shafarevich groups of elliptic curves under prime quadratic twists.