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

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 $|k-K|<K^{\theta }$ 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 $\delta <1/2$.