Selberg orthogonality for half-integral weight modular forms

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{łł}'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: ${\mathrm{G}\mathrm{L}}_2$ 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.