The geometric and arithmetic volume of Shimura varieties of orthogonal type

TL;DR

This paper calculates the arithmetic volumes of orthogonal type Shimura varieties using Borcherds products, integral models, and automorphic vector bundles, providing evidence for Kudla's conjectures relating heights and Eisenstein series.

Contribution

It introduces a method to compute arithmetic volumes of Shimura varieties of orthogonal type leveraging Borcherds products and integral models, advancing understanding of their arithmetic geometry.

Findings

Arithmetic volumes computed up to contributions from very bad primes.

Evidence supporting Kudla's conjectures on heights and derivatives of Eisenstein series.

Development of a functorial theory of integral models and automorphic vector bundles.

Abstract

We apply the theory of Borcherds products to calculate arithmetic volumes (heights) of Shimura varieties of orthogonal type up to contributions from very bad primes. The approach is analogous to the well-known computation of their geometric volume by induction, using special cycles. A functorial theory of integral models of toroidal compactifications of those varieties and a theory of arithmetic Chern classes of integral automorphic vector bundles with singular metrics are used. We obtain some evidence in the direction of Kudla's conjectures on relations of heights of special cycles on these varieties to special derivatives of Eisenstein series.