Cancellation in Sums of Hecke Eigenvalues Over Quadratic Polynomials and Mass Equidistribution

TL;DR

This paper demonstrates square root cancellation in sums of Hecke eigenvalues over quadratic polynomials, relates it to mass equidistribution of modular forms, and shows that almost all forms satisfy this conjecture with an effective rate.

Contribution

It establishes square root cancellation in sums of Hecke eigenvalues over quadratic polynomials and connects this to mass equidistribution, providing effective convergence rates for almost all forms.

Findings

Square root cancellation in sums over quadratic polynomials.

Almost all forms satisfy the mass equidistribution conjecture.

Effective convergence rate of $k^{- ext{delta}}$ for forms.

Abstract

We study cancellation in sums of Hecke eigenvalues over irreducible quadratic polynomials over short intervals. In particular, we look at an average over bases of Hecke forms of weight $k$ in the range $\vert k-K\vert<K^\theta$ where $1/3<\theta<1$. We see that when averaged over this family such sums admit square root cancellation. The key new arithmetic input for such a result is a bound on sums of Kloosterman sums over irreducible quadratic polynomials. Then using work of Nelson, we relate such sums to the mass equidistribution conjecture for modular forms on compact arithmetic surfaces, and we show that almost all forms satisfy the mass equidistribution conjecture. Furthermore, such forms will satisfy the conjecture with an effective convergence rate of $k^{-\delta}$ for any $\delta<1/2$.