Automorphic Forms and Arithmetic

Spherical varieties play an important role in the study of periods of automorphic forms. But very closely related varieties can lead to very distinct arithmetic problems. Motivated by applications to relative trace formulas, we discuss the natural question of distinguishing different forms of a given spherical variety in arithmetic settings, giving a solution for symmetric varieties. It turns out that the answer is intimately connected with the construction of the dual Hamiltonian variety associated with the symmetric variety by Ben-Zvi, Sakellaridis, and Venkatesh. I will explain the source of these questions in the theory of endoscopy for symmetric varieties, with application to the (pre-)stabilization of relative trace formulas.

Arithmetic Siegel--Weil formulas predict that derivatives of Eisenstein series Fourier coefficients should encode arithmetic volumes of special cycles on Shimura varieties. Such formulas are arithmetic versions of formulas for complex volumes of moduli spaces in terms of L-functions and Eisenstein series. These "geometrize" the classical Siegel mass and Siegel--Weil formulas, on lattice and lattice vector counting. I will introduce these ideas and overview my proof for arithmetic 1-cycles on unitary Shimura varieties of arbitrarily high dimension. The key input is a new "limit phenomenon" relating positive characteristic intersection numbers and heights in mixed characteristic. I will also mention a closely related "near-center arithmetic fundamental lemma (AFL)" relating arithmetic 1-cycles and orbital integrals. This "AFL" is from joint work in progress with Weixiao Lu and Wei Zhang.