Simultaneous equidistribution of toric periods and fractional moments of \(L\)-functions (Q6566454)

The problem of equidistribution of integer points on the sphere admits many variants, such as the distribution of Heegner points on the modular curve, packets of closed geodesics on the modular curve, as well as the supersingular reduction of CM elliptic curves. In this paper, the authors review these classical examples. In each case, the underlying set of arithmetic objects admits an action by the class group of a quadratic order. Furthermore, they have in common the equidistribution of the adelic quotient \([\mathrm{\mathbf{{T}}}]= \mathrm{\mathbf{{T}}}(\mathbb{Q})\backslash \mathrm{\mathbf{{T}}}(\mathbb{A})\) of an algebraic torus \(\mathbf{T}\) of large discriminant inside the automorphic space \([\mathrm{\mathbf{{G}}}]= \mathrm{\mathbf{{G}}}(\mathbb{Q})\backslash \mathrm{\mathbf{{G}}}(\mathbb{A})\) of an inner form \(\mathrm{\mathbf{{G}}}\) of \(\mathrm{PGL}_2\). The authors consider a generalization of Duke's theorem by taking a torus embedded diagonally into two distinct inner forms of \(\mathrm{PGL}_2\) and they establish the conjecture of \textit{P. Michel} and \textit{A. Venkatesh} for diagonal packets [in: Proceedings of the international congress of mathematicians (ICM), Madrid, Spain, 2006. Volume II: Invited lectures. Zürich: European Mathematical Society (EMS). 421--457 (2006; Zbl 1157.11019)], under the assumption of the generalized Riemann hypothesis (GRH). The main tool of the proof relies on an estimate of the fractional moments of a certain family of Rankin-Selberg \(L\)-functions.