A relative trace formula for a compact Riemann surface

TL;DR

This paper develops a relative trace formula for compact Riemann surfaces with respect to a closed geodesic, linking period and ortholength spectra, and deriving asymptotic and nonvanishing results for geometric and spectral data.

Contribution

It introduces a new relative trace formula relating periods and ortholengths on Riemann surfaces, providing novel proofs and nonvanishing results.

Findings

Asymptotic estimates for periods of Laplacian eigenforms along geodesics

Bounds on lengths of orthogonal geodesic segments

Nonvanishing results for multiple periods

Abstract

We study a relative trace formula for a compact Riemann surface with respect to a closed geodesic $C$. This can be expressed as a relation between the period spectrum and the ortholength spectrum of $C$. This provides a new proof of asymptotic results for both the periods of Laplacian eigenforms along $C$ as well estimates on the lengths of geodesic segments which start and end orthogonally on $C$. Variant trace formulas also lead to several simultaneous nonvanishing results for different periods.