Decorrelation phenomena in mixed moments for orthogonal families of $L$-Functions and their applications: automorphic periods, Fourier coefficients of half-integral weight forms, and isotropy subgroups of Tate--Shafarevich groups

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 Radziwiłł'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. Under the assumptions of the GRH and the Generalized Ramanujan Conjecture, we prove upper bound estimates for their moments and present the following three arithmetic applications: i) The decorrelation of automorphic periods averaged over imaginary quadratic fields. ii) The decorrelation of Fourier coefficients of half-integral weight modular forms. iii) The decorrelation of the analytic orders of isotropy subgroups of Tate--Shafarevich groups of elliptic curves under quadratic twists.