Joint distribution of Hecke eigenforms

TL;DR

This paper investigates the joint distribution of Hecke eigenforms, proving asymptotic formulas and decorrelation results under GRH and GRC, with implications for triple product L-functions.

Contribution

It formulates conjectures on joint distributions and proves asymptotic formulas and decorrelation results under GRH and GRC.

Findings

Asymptotic formula for joint mass of two Hecke eigenforms

Higher decorrelation of eigenforms asymptotically vanishes under GRH

Asymptotic formula for the first moment of triple product L-functions

Abstract

In this paper, we formulate conjectures on the joint distribution of several Hecke eigenforms. We prove an asymptotic formula of the joint mass of two Hecke eigenforms under the generalized Riemann Hypothesis (GRH) and the generalized Ramanujan conjecture (GRC). We also show that a higher decorrelation of two Hecke eigenforms asymptotically vanishes under GRH. As a consequence, we prove an asymptotic formula for the first moment of the triple product $L$-functions under GRH and GRC.