Abstract:

Theta series for lattices count lattice vectors of fixed norm. Such theta series give some of the first examples of automorphic forms.

It is possible to form "theta series" in other geometric contexts, e.g. for counting problems involving abelian varieties.
It is expected that these theta series again have additional automorphic symmetry.

I will explain some “near-central” instances of an arithmetic Siegel--Weil formula from Kudla’s program. These "geometrize" the classical Siegel--Weil formulas, on lattice and lattice vector counting via Eisenstein series.

At these near-central points of functional symmetry, we observe that both the "leading" special value (complex volumes) and the "subleading" first derivative (arithmetic volume) simultaneously have geometric meaning.

The key input is a new "limit phenomenon" relating positive characteristic intersection numbers and heights in mixed characteristic, as well as its automorphic counterpart.