The L² restriction of the Eisenstein series to a geodesic segment

2015/08/06 by Matthew P. Young, Young, Matthew P.
Mathematics · #11F03 #11M41 #58J51 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analysis of PDEs (math.AP) #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1508.01470

openalex publication_date 2015/08/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the L² norm of the Eisenstein series E(z,1/2+iT) restricted to a segment of a geodesic connecting infinity and an arbitrary real. We conjecture that on slightly thickened geodesics of this form, the Eisenstein series satifies restricted QUE. We prove a lower bound that matches this predicted asymptotic. We also prove an upper bound that nearly matches the lower bound assuming the Riemann Hypothesis (unconditionally, the sharp upper bound holds for almost all T). Finally, we show the restricted QUE conjecture for geodesics with rational endpoints.

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