Picard rank jumps for K3 surfaces with bad reduction

Tayou, Salim

doi:10.2140/ant.2025.19.77

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Picard rank jumps for K3 surfaces with bad reduction

Salim Tayou

Vol. 19 (2025), No. 1, 77–112

DOI: 10.2140/ant.2025.19.77

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 and Tang to the case where $X$ has bad reduction. We prove a similar result for generically ordinary nonisotrivial families of K3 surfaces over curves over ${\bar{𝔽}}_{p}$ which extends previous work of Maulik, Shankar and Tang. As a consequence, we give a new proof of the ordinary Hecke orbit conjecture for orthogonal and unitary Shimura varieties.

Keywords

K3 surfaces, GSpin Shimura varieties, Arakelov theory

Mathematical Subject Classification

Primary: 11G18, 14G40, 14J28

Milestones

Received: 27 June 2022

Revised: 31 July 2023

Accepted: 9 January 2024

Published: 4 December 2024

Authors

Salim Tayou
Department of Mathematics Harvard University Cambridge, MA United States

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Vol. 19, No. 1, 2025