Picard rank jumps for K3 surfaces with bad reduction

Abstract

Let X be a K3 surface over a number field. We prove that X has infinitely many specializations where its Picard rank jumps, hence extending our previous work with Shankar-Shankar-Tang to the case where X might have potentially bad reduction. We prove a similar result for generically ordinary non-isotrivial families of K3 surfaces over curves over F p which extends previous work of Maulik-Shankar-Tang. As a consequence, we give a new proof of the ordinary Hecke orbit conjecture for orthogonal and unitary Shimura varieties.

Rational curves on elliptic K3 surfaces

“…Let Υ be a toroidal stratum representative of type II , B Υ the corresponding boundary component of type II and we assume in this section that that boundary point S (F P ) lies in B Υ (F p ). Then by [Tay22], we have Combining Equation (6.2.1) and the previous estimate, we get Proposition 3.9 in the type II case. Combining the two previous estimates concludes the proof of Proposition 3.9 in the type III case.…”

“…We introduce in this section Arakelov Chow groups following [GS90] and [BBK07]. For more details on this section, we also refer to [SSTT22, §3.1] and [Tay22,§3]. Let (rL, Q) ⊂ (L, Q) be an inclusion of quadratic lattices of signature (n, 2) as before, in particular L is maximal with discriminant coprime to r. Let M r , M be the normal integral models over Z of the GSpin Shimura varieties associated to (rL, Q) and (L, Q) constructed in the previous section.…”

“…Let P be a prime of bad reduction. The toroidal compactification M Σ r has a stratification with two types of boundary components as explained in [Tay22]. We will use the results from that paper to analyze the local intersection multiplicities and we focus now on boundary components of type II.…”

Exceptional jumps of Picard ranks of reductions of K3 surfaces over number fields

“…We should mention that the assumption of potentially good reduction in the above statements has been removed recently by one of the authors in [Tay22] so that all the above results hold unconditionally.…”

“…Indeed, our idea is a crucial ingredient in proving intersection-theoretic results in characteristic p, as well as in proving the ordinary Hecke-orbit conjecture for GSpin Shimura varieties (see [MST22]). It has also been used to extend the main theorem to the bad reduction situation; see [Tay22]. We note, however, that our theorem applies to K3 surfaces with potentially good reduction everywhere.…”