Spectral $L^2$-bounds on Eisenstein series and Diophantine properties of quadratic forms

Abstract

We give $L^2$-bounds with respect to the height of the spectral parameter for Epstein zeta functions $\mathrm{Z}_Q(s)$ and non-holomorphic Eisenstein series $E_{\mathfrak{c}}(s,z)$ attached to the Hecke congruence subgroup $\Gamma_0 (p)$ of prime level $p$. Our bounds are completely explicit in the level and, additionally, constitute an almost sure logarithmic improvement over previously established bounds for the level-one Eisenstein series. Following the approach of Kelmer--Kontorovich--Lutsko, we combine the approximate functional equation for $\mathrm{Z}_Q(s)$ with estimates for the number of pairs of lattice vectors at which $Q$ takes nearly equal values. Under a Diophantine hypothesis on $Q$, we sharpen these counting estimates using results of Eskin--Margulis--Mozes on the distribution of values of quadratic forms of signature $(2,2)$.