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.

A uniform bound for geodesic periods of eigenfunctions on hyperbolic surfaces

“…1. Periods along closed geodesics have been studied over years from different perspectives (see e.g., [9,20,24,28,38]). i / " i to hold for any " > 0.…”

The hyperbolic lattice point problem in conjugacy classes

“…The exact asymptotic behavior was first proved by Good [11, Theorem 2, p. 108], see also [17] and in bigger generality by Tsuzuki [23, Theorem 1, p. 2].…”

A symplectic restriction problem

“…With applications to the above mentioned symplectic restriction problem in mind, we are interested in pairs of Heegner periods in the spectral average ev P(D 1 ; u)P(D 2 ; u)h(t u )du for a suitable test function h, two discriminants D 1 , D 2 < 0 and ev as in Proposition 1. While pairs of geodesics have been studied in a few situations [63][64][65] (but only for compact Riemann surfaces and not from an arithmetic point of view), to the best of our knowledge nothing seems to be known about spectral averages of pairs of Heegner periods. Opening the sums in the definition of P(D 1 ; u) and P(D 2 ; u), this can be expressed as a double sum of an automorphic kernel…”