We define variants of PEL type of the Shimura varieties that appear in the context of the arithmetic Gan–Gross–Prasad (AGGP) conjecture. We formulate for them a version of the AGGP conjecture. We also construct (global and semi-global) integral models of these Shimura varieties and formulate for them conjectures on arithmetic intersection numbers. We prove some of these conjectures in low dimension.

In this paper, we prove a conjecture of Wei Zhang on comparison of certain local relative characters from which we draw some consequences for the Ichino–Ikeda conjecture for unitary groups.

For the group $G=\operatorname{PGL}_{2}$ we perform a comparison between two relative trace formulas: on the one hand, the relative trace formula of Jacquet for the quotient $T\backslash G/T$, where $T$ is a nontrivial torus, and on the other the Kuznetsov trace formula (involving Whittaker periods), applied to nonstandard test functions. This gives a new proof of the celebrated result of Waldspurger on toric periods, and suggests a new way of comparing trace formulas, with some analogies to Langlands’ ‘Beyond Endoscopy’ program.

We use a relative trace formula on $\text{GL}(2)$ to compute a sum of twisted modular $L$-functions anywhere in the critical strip, weighted by a Fourier coefficient and a Hecke eigenvalue. When the weight $k$ or level $N$ is sufficiently large, the sum is nonzero. Specializing to the central point, we show in some cases that the resulting bound for the average is as good as that predicted by the Lindelöf hypothesis in the $k$ and $N$ aspects.

We establish the existence of smooth transfer for Guo–Jacquet relative trace formulae in the $p$-adic case. This kind of smooth transfer is a key step towards a generalization of Waldspurger’s result on central values of L-functions of $\text{GL}_{2}$.

Let $E$ be a $\textrm{CM}$-field and $\pi$ a cuspidal representation of $\textrm{GL}_n({\mathbb{A}}_E)$ which admits a spherical vector (at all places) <formula form="inline" disc="math" id="ffm005"><formtex notation="AMSTeX">$\phi_0$. We evaluate the period of $\phi_0$ with respect to any compact unitary group. The result is consistent with a conjecture of Sarnak.

The relative trace formula is a tool in the theory of automorphic forms which was invented by Jacquet in order to study period integrals and relate them to Langlands functoriality. In this paper we give an analogue of Arthur’s spectral expansion of the trace formula to the relative setup in the context of $\text{GL}_n$. This is an important step toward application of the relative trace formula and it extends earlier work by several authors to higher rank. Our method is new and based on complex analysis and majorization of Eisenstein series. To that end we use recent lower bounds of Brumley for Rankin–Selberg $L$-functions at the edge of the critical strip.

In this paper we show how to predict relative trace identities from the computation of Jacquet modules of the Weil representations. Many previously considered special cases of relative trace identities fit the principle we develop here, including those with important applications on L-functions. We also show how to prove these identities using the Weil representation. We give a proof of the relative trace identities between the distributions on SO(n + 1, n) and $\widetilde{\textit{Sp}}(m)$$(n\geq m)$. The proof should serve as a model to the other cases conjectured in the paper.