Twisted relative trace formulae with a view towards unitary groups

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 $\langle\sigma\rangle={\rm Gal}(E/F)$, let $U^{\sigma}$ be a quasi-split unitary group with respect to a CM extension $M/F$, and let $U:={\rm Res}_{E/F}U^{\sigma}$. Under suitable local hypotheses, we show that a cuspidal cohomological automorphic representation $\pi$ 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 $\pi'$ of $U$ that is $U^{\sigma}$-distinguished in the sense that there is a form in the space of $\pi'$ admitting a nonzero period over $U^{\sigma}$. 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.