The Geometric and Arithmetic Volume of Shimura Varieties of Orthogonal Type

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.

Faltings heights of abelian varieties with complex multiplication

“…is a weakly holomorphic form, so that f = f + and c f (m) = c + f (m). The following result will be shown in the companion paper [HM15], generalizing a result of F. Hörmann [Hör14]. Here, we only sketch its proof.…”

“…Here, we only sketch its proof. For the applications to Colmez's conjecture, we will only require the assertion over primes of good reduction, which is already contained in [Hör14].…”

“…In particular, Hypothesis 8.3.1 is satisfied, and, since L ◇ (p) is self-dual, by Theorem 4.8.1, E ◇ (f ) does not intersect M ◇ Fp , as desired. We note again that the proof only used knowledge of the divisor of the Borcherds lift of f at primes where L ◇ is self-dual, which is contained in [Hör14], and not the full strength of Theorem 4.8.1.…”

“…Combining Theorem B with (1.3.2) gives For simplicity, assume that d ≥ 4 (this guarantees that V contains an isotropic line; throughout the body of the paper, we only require d ≥ 2). After possibly replacing f by a positive integer multiple, the theory of Borcherds products [Bor98, Hör14,HM15] gives us a rational section Ψ(f ) of the line bundle ω ⊗c f (0,0) , satisfying − log Ψ(f ) 2 = Φ(f ) − c f (0, 0) log(4πe γ ), and satisfying div(Ψ(f )) = Z(f ) up to a linear combination of irreducible components of the special fiber M F 2 . We define a Cartier divisor E 2 (f ) = div(Ψ(f )) − Z(f ), on M, supported entirely in characteristic 2, which should be viewed as an unwanted error term.…”

Green forms and the arithmetic Siegel–Weil formula

“…A slightly weaker version of this conjecture (i.e. away from an explicit set of primes determined by T and ϕ) was proved for general orthogonal Shimura varieties over Q by Hörmann [15]. Several cases involving cycles of top arithmetic codimension supported at finite primes were also established by Kudla and Rapoport, see e.g.…”