A non-split sum of coefficients of modular forms

TL;DR

This paper investigates truncated sums of Hecke eigenvalues for $GL_2$ automorphic forms along quadratic polynomials, establishing power savings and applying results to moments of $L$-values, Heegner points, and singular moduli.

Contribution

It introduces new truncated sums of Hecke eigenvalues along quadratic polynomials and derives power-saving estimates with applications to $L$-values and algebraic points.

Findings

Established power-saving estimates for truncated sums.

Applied results to moments of critical $L$-values.

Analyzed asymptotic behavior of Heegner points and singular moduli.

Abstract

We shall introduce and study certain truncated sums of Hecke eigenvalues of $GL_2$-automorphic forms along quadratic polynomials. A power saving estimate is established and new applications to moments of critical $L$-values associated to quadratic fields are derived. An application to the asymptotic behavior of the height of Heegner points and singular moduli is discussed in details.