A relative trace formula for a compact Riemann surface

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.

35 pages. This version contains minor corrections to the published version. The only change to the main results is a modification of the constants in the final theorem and corollary