A nonsplit sum of coefficients of modular forms

Abstract

We introduce and study certain truncated sums of Hecke eigenvalues of ${\mathrm{GL}}_2$-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 detail.