Arithmetic volumes of unitary Shimura varieties

TL;DR

This paper computes the arithmetic volumes of certain line bundles and divisors on unitary Shimura varieties, linking them to special values of L-functions and Eisenstein series, advancing understanding in arithmetic geometry.

Contribution

It provides explicit formulas for arithmetic volumes of line bundles and divisors on unitary Shimura varieties, connecting geometric invariants to automorphic forms and L-functions.

Findings

Arithmetic volume expressed via derivatives of Dirichlet L-functions

Volumes of Kudla-Rapoport divisors related to Eisenstein series coefficients

Explicit formulas for metrized line bundles on Shimura varieties

Abstract

The integral model of a $\mathrm{GU}(n-1,1)$ Shimura variety carries a universal abelian scheme over it, and the dual top exterior power of its Lie algebra carries a natural hermitian metric. We express the arithmetic volume of this metrized line bundle, defined as an iterated self-intersection in the arithmetic Chow ring, in terms of logarithmic derivatives of Dirichlet $L$-functions. We also determine the arithmetic volumes of Kudla-Rapoport divisors and relate them to coefficients of Eisenstein series.