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arXiv · 2610.37420

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

Jitendra Bajpai·Kristian Holm
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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 $Γ_0 (p)$ 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 $(2,2)$.

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Jitendra Bajpai, Kristian Holm. 2026-10-27. Spectral $L^2$-bounds on Eisenstein series and Diophantine properties of quadratic forms. https://arxiv.org/abs/2610.37420

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