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

Ananth N. Shankar; Arul Shankar; Yunqing Tang; Salim Tayou

doi:10.1017/fmp.2022.14

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

Ananth N. Shankar, Arul Shankar, Yunqing Tang, Salim Tayou

Mathematics

Research output: Contribution to journal › Article › peer-review

8 Scopus citations

Abstract

Given a K3 surface X over a number field K with potentially good reduction everywhere, we prove that the set of primes of K where the geometric Picard rank jumps is infinite. As a corollary, we prove that either has infinitely many rational curves or X has infinitely many unirational specialisations. Our result on Picard ranks is a special case of more general results on exceptional classes for K3 type motives associated to GSpin Shimura varieties. These general results have several other applications. For instance, we prove that an abelian surface over a number field K with potentially good reduction everywhere is isogenous to a product of elliptic curves modulo infinitely many primes of K.

Original language	English (US)
Article number	e21
Journal	Forum of Mathematics, Pi
Volume	10
DOIs	https://doi.org/10.1017/fmp.2022.14
State	Published - Sep 26 2022

Funding

A.N.S. is partially supported by the NSF grant DMS-2100436. A.S. is supported by an NSERC Discovery grant and a Sloan fellowship. Y.T. is partially supported by the NSF grant DMS-1801237. S.T. has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 715747).

ASJC Scopus subject areas

Analysis
Algebra and Number Theory
Statistics and Probability
Mathematical Physics
Geometry and Topology
Discrete Mathematics and Combinatorics

Access to Document

10.1017/fmp.2022.14

Cite this

@article{9c05c6fa752d47ff84d797fbb86d9040,

title = "Exceptional jumps of Picard ranks of reductions of K3 surfaces over number fields",

abstract = "Given a K3 surface X over a number field K with potentially good reduction everywhere, we prove that the set of primes of K where the geometric Picard rank jumps is infinite. As a corollary, we prove that either has infinitely many rational curves or X has infinitely many unirational specialisations. Our result on Picard ranks is a special case of more general results on exceptional classes for K3 type motives associated to GSpin Shimura varieties. These general results have several other applications. For instance, we prove that an abelian surface over a number field K with potentially good reduction everywhere is isogenous to a product of elliptic curves modulo infinitely many primes of K.",

author = "Shankar, {Ananth N.} and Arul Shankar and Yunqing Tang and Salim Tayou",

note = "Funding Information: A.N.S. is partially supported by the NSF grant DMS-2100436. A.S. is supported by an NSERC Discovery grant and a Sloan fellowship. Y.T. is partially supported by the NSF grant DMS-1801237. S.T. has received funding from the European Research Council (ERC) under the European Union{\textquoteright}s Horizon 2020 research and innovation programme (grant agreement No 715747). Publisher Copyright: {\textcopyright} The Author(s), 2022. Published by Cambridge University Press.",

year = "2022",

month = sep,

day = "26",

doi = "10.1017/fmp.2022.14",

language = "English (US)",

volume = "10",

journal = "Forum of Mathematics, Pi",

issn = "2050-5086",

publisher = "Cambridge University Press",

}

TY - JOUR

T1 - Exceptional jumps of Picard ranks of reductions of K3 surfaces over number fields

AU - Shankar, Ananth N.

AU - Shankar, Arul

AU - Tang, Yunqing

AU - Tayou, Salim

N1 - Funding Information: A.N.S. is partially supported by the NSF grant DMS-2100436. A.S. is supported by an NSERC Discovery grant and a Sloan fellowship. Y.T. is partially supported by the NSF grant DMS-1801237. S.T. has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 715747). Publisher Copyright: © The Author(s), 2022. Published by Cambridge University Press.

PY - 2022/9/26

Y1 - 2022/9/26

N2 - Given a K3 surface X over a number field K with potentially good reduction everywhere, we prove that the set of primes of K where the geometric Picard rank jumps is infinite. As a corollary, we prove that either has infinitely many rational curves or X has infinitely many unirational specialisations. Our result on Picard ranks is a special case of more general results on exceptional classes for K3 type motives associated to GSpin Shimura varieties. These general results have several other applications. For instance, we prove that an abelian surface over a number field K with potentially good reduction everywhere is isogenous to a product of elliptic curves modulo infinitely many primes of K.

AB - Given a K3 surface X over a number field K with potentially good reduction everywhere, we prove that the set of primes of K where the geometric Picard rank jumps is infinite. As a corollary, we prove that either has infinitely many rational curves or X has infinitely many unirational specialisations. Our result on Picard ranks is a special case of more general results on exceptional classes for K3 type motives associated to GSpin Shimura varieties. These general results have several other applications. For instance, we prove that an abelian surface over a number field K with potentially good reduction everywhere is isogenous to a product of elliptic curves modulo infinitely many primes of K.

UR - http://www.scopus.com/inward/record.url?scp=85138864780&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85138864780&partnerID=8YFLogxK

U2 - 10.1017/fmp.2022.14

DO - 10.1017/fmp.2022.14

M3 - Article

AN - SCOPUS:85138864780

SN - 2050-5086

VL - 10

JO - Forum of Mathematics, Pi

JF - Forum of Mathematics, Pi

M1 - e21

ER -

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

Abstract

Funding

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this