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.

We introduce a twisted relative trace formula which simultaneously generalizes the twisted trace formula of Langlands et. al. (in the quadratic case) and the relative trace formula of Jacquet and Lai. Certain matching statements relating this twisted relative trace formula to a relative trace formula are also proven (including the relevant fundamental lemma in the "biquadratic case"). Using recent work of Jacquet, Lapid and their collaborators and the Rankin-Selberg integral representation of the Asai L-function (obtained by Flicker using the theory of Jacquet, Piatetskii-Shapiro, and Shalika), we give the following application: Let E/F be a totally real quadratic extension with σ = Gal(E/F ), let U σ be a quasi-split unitary group with respect to a CM extension M/F , and let U := Res E/F U σ . Under suitable local hypotheses, we show that a cuspidal cohomological automorphic representation π of U whose Asai L-function has a pole at the edge of the critical strip is nearly equivalent to a cuspidal cohomological automorphic representation π of U that is U σ -distinguished in the sense that there is a form in the space of π admitting a nonzero period over U σ . This provides cohomologically nontrivial cycles of middle dimension on unitary Shimura varieties analogous to those on Hilbert modular surfaces studied by Harder, Langlands, and Rapoport.

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Abstract. Gelbart and Piatetskii-Shapiro constructed various integral representations of Rankin-Selberg type for groups G × GLn, where G is of split rank n. Here we show that their method can equally well be applied to the product U 3 × GL 2 , where U 3 denotes the quasisplit unitary group in three variables. As an application, we describe which cuspidal automorphic representations of U 3 occur in the Siegel induced residual spectrum of the quasisplit U 4 .