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.

Following Jacquet, Lapid and Rogawski, we regularize trilinear periods. We use the regularized trilinear periods to compute Fourier–Jacobi periods of residues of Eisenstein series on metaplectic groups, which has an application to the Gan–Gross–Prasad conjecture.

Beilinson [Higher regulators and values of L-functions, Itogi Nauki i Tekhniki Seriya Sovremennye Problemy Matematiki Noveishie Dostizheniya (Current problems in mathematics), vol. 24 (Vserossiisky Institut Nauchnoi i Tekhnicheskoi Informatsii, Moscow, 1984), 181–238] obtained a formula relating the special value of the L-function of H2 of a product of modular curves to the regulator of an element of a motivic cohomology group, thus providing evidence for his general conjectures on special values of L-functions. In this paper we prove a similar formula for the L-function of the product of two Drinfeld modular curves, providing evidence for an analogous conjecture in the case of function fields.

We show that the extrapolation to the case of global fields of characteristic p of a question posed by Stark in 1980, regarding abelian L-functions of order of vanishing 2 at s = 0, has a negative answer. We provide links between various versions of Stark's question and a natural refinement of Brumer's conjecture, in the general context of global fields of arbitrary characteristic. As a consequence, we show that the refinement of Brumer's conjecture is, in general, false for characteristic p global fields.