Special cycles on toroidal compactifications of orthogonal Shimura varieties

Abstract

We determine the behavior of automorphic Green functions along the boundary components of toroidal compactifications of orthogonal Shimura varieties. We use this analysis to define boundary components of special divisors and prove that the generating series of the resulting special divisors on a toroidal compactification is modular.

Picard rank jumps for K3 surfaces with bad reduction

“…In this section, we describe the toroidal compactifcations of M as well as the structure of the boundary components following [HZ21] and [HP20]. See also [AMRT10] for the general theory of toroidal compactifications over C.…”

“…We follow a similar approach for Theorem 1.3, using the usual intersection theory on the reduction modulo p of the aforementioned compactification of the integral model of a GSpin Shimura variety. The new ingredients which were missing in both [SSTT19] and [MST22] are the local estimates on multiplicities of intersection with special divisors at points of bad reduction and the estimates of extra terms coming from the boundary divisors in the global intersection numbers coming from the work of [HZ21]. These are the main contributions of this paper.…”

“…We give an estimate on the growth of the successive minima of these lattices, then a geometry-of-numbers type argument allows us to derive the desired estimates. To obtain the bounds on the extra terms in the global intersection number, we use the explicit expressions from [HZ21] and [Bru02] combined with an equidistribution result from [Duk88,EO06].…”

“…If P is the stabilizer of a primitive isotropic plane J Q ⊂ L Q , then Φ is said to be of type III. We will follow the notations of [HZ21] and denote by Υ, resp. Ξ, a cusp label representative of type II, resp.…”

“…Then P is the stabilizer of a primitive isotropic plane Υ is a modular curve and the abelian scheme A Υ is equal to the Kuga-Sato variety D ⊗ E where E → M h Υ is the universal elliptic curve over over M h Υ and D is the positive definite plane J ⊥ /J, see [HZ21, Corollary 3.17] and [Zem20, Proposition 4.3] for details and proofs. Notice that our choice for the compact open subgroup K gives exactly the stable orthogonal group used in [HZ21] and [Zem20].…”