• Corpus ID: 202583501

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

@article{Shankar2019ExceptionalJO,
title={Exceptional jumps of Picard ranks of reductions of K3 surfaces over number fields},
author={Ananth N. Shankar and Arul Shankar and Yunqing Tang and Salim Tayou},
journal={arXiv: Number Theory},
year={2019}
}
• Published 17 September 2019
• Mathematics
• arXiv: Number Theory
Given a K3 surface $X$ over a number field $K$, we prove that the set of primes of $K$ where the geometric Picard rank jumps is infinite, assuming that $X$ has everywhere potentially good reduction. The result is a special case of a more general one on exceptional classes for K3 type motives associated to GSpin Shimura varieties and several other applications are given. As a corollary, we give a new proof of the fact that $X_{\overline{K}}$ has infinitely many rational curves.
7 Citations
Picard rank jumps for K3 surfaces with bad reduction
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
Reduction of Brauer classes on K3 surfaces, rationality and derived equivalence
• Mathematics
• 2021
Given a smooth projective variety over a number eld and an element of its Brauer group, we consider the specialization of the Brauer class at a place of good reduction for the variety and the class.
Curves on K3 surfaces
• Mathematics
• 2019
We show that every projective K3 surface over an algebraically closed field of characteristic zero contains infinitely many rational curves. For this, we introduce two new techniques in the
Picard ranks of K3 surfaces over function fields and the Hecke orbit conjecture
• Mathematics
Inventiones mathematicae
• 2022
Let $\mathscr{X} \rightarrow C$ be a non-isotrivial and generically ordinary family of K3 surfaces over a proper curve $C$ in characteristic $p \geq 5$. We prove that the geometric Picard rank jumps
Equidistribution of Hodge loci II
• Mathematics
• 2021
Let H ⊂ G be semisimple Lie groups, Γ ⊂ G a lattice and K a compact subgroup of G. For n ∈ N, let On be the projection to Γ\G/K of a finite union of closed H-orbits in Γ\G. In this very general
From sum of two squares to arithmetic Siegel-Weil formulas
The main goal of this expository article is to survey recent progress on the arithmetic Siegel–Weil formula and its applications. We begin with the classical sum of two squares problem and put it in
Unlikely and just likely intersections for high dimensional families of elliptic curves
Given two varieties V , W in the n-fold product of modular curves, we answer afﬁrmatively a question (formulated by Shou-Wu Zhang’s AIM group) on whether the set of points in V that are Hecke

## References

SHOWING 1-10 OF 107 REFERENCES
On the distribution of the Picard ranks of the reductions of a K3 surface
• Mathematics
• 2016
We report on our results concerning the distribution of the geometric Picard ranks of $K3$ surfaces under reduction modulo various primes. In the situation that $\rk \Pic S_{\overline{K}}$ is even,
On the Picard number of K3 surfaces over number fields
We discuss some aspects of the behavior of specialization at a finite place of Neron-Severi groups of K3 surfaces over number fields. We give optimal lower bounds for the Picard number of such
Tate's conjecture for K3 surfaces of finite height
• Mathematics
• 1985
This paper extends the proof [16] of the Tate conjecture for ordinary K3 surfaces over a finite field to the more general case of all K3's of finite height. As in [16], our method is to find a
Curves on K3 surfaces
• Mathematics
• 2019
We show that every projective K3 surface over an algebraically closed field of characteristic zero contains infinitely many rational curves. For this, we introduce two new techniques in the
Supersingular K3 surfaces for large primes
Given a K3 surface X over a field of characteristic p, Artin conjectured that if X is supersingular (meaning infinite height) then its Picard rank is 22. Along with work of Nygaard-Ogus, this
Reductions of abelian surfaces over global function fields
• Mathematics
Compositio Mathematica
• 2022
Let $A$ be a non-isotrivial ordinary abelian surface over a global function field of characteristic $p>0$ with good reduction everywhere. Suppose that $A$ does not have real multiplication by any
Height pairings on orthogonal Shimura varieties
• Mathematics
Compositio Mathematica
• 2017
Let $M$ be the Shimura variety associated to the group of spinor similitudes of a quadratic space over $\mathbb{Q}$ of signature $(n,2)$ . We prove a conjecture of Bruinier and Yang, relating the
Faltings heights of CM cycles and derivatives of L-functions
• Mathematics
• 2008
We study the Faltings height pairing of arithmetic special divisors and CM cycles on Shimura varieties associated to orthogonal groups. We compute the Archimedean contribution to the height pairing
Exceptional splitting of reductions of abelian surfaces
• Mathematics
• 2017
Heuristics based on the Sato--Tate conjecture suggest that an abelian surface defined over a number field has infinitely many places of split reduction. We prove this result for abelian surfaces