The Relative Trace Formula for Galois Periods

Let $E\mathrm{/}F$ be a quadratic extension of number fields. We introduce truncated geometric and spectral RTF distributions associated to a Galois symmetric pair $G\subset {\mathrm{R}\mathrm{e}\mathrm{s}}_{E\mathrm{/}F}{G}_E$, subject to the constraint that $G$ and ${\mathrm{R}\mathrm{e}\mathrm{s}}_{E\mathrm{/}F}{G}_E$ have the same split rank, and formulate a precise coarse RTF identity. Specializing to $S{L}_{2,F}\subset {\mathrm{R}\mathrm{e}\mathrm{s}}_{E\mathrm{/}F}S{L}_{2,E}$, we show that the truncated geometric RTF distribution converges, and is given by a linear polynomial in the truncation parameter. We then compute the fine geometric expansion explicitly, including the contribution of the regularized relative unipotent orbital integrals. We propose a geometric viewpoint which guided the computation of these unipotent terms.