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

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

doi:10.48550/arxiv.1909.07473

Cited by 2 publications

(2 citation statements)

References 32 publications

(54 reference statements)

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“…In light of Example 1.3, it would be very interesting to show that, if one relaxes the requirement that S(X, α) contain a set of positive natural density to the weaker statement that it is infinite, then Theorem 1.1 holds when E is a CM field or dim E (T(X) Q ) is even. Recent work of Shankar, Shankar, Tang and Tayou [SSTT19] suggests that such a statement may be within reach.…”

Section: Introductionmentioning

confidence: 99%

Reduction of Brauer classes on K3 surfaces, rationality and derived equivalence

Frei¹,

Hassett²,

Várilly-Alvarado³

2021

Preprint

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We consider the reduction of Brauer classes on surfaces over number fields, with a view toward applications to rationality and derived equivalence. We show that a Brauer class on a very general polarized K3 surface over a number field becomes trivial upon reduction for a set of places of positive natural density. As a consequence, there are cubic fourfolds which become rational upon reduction for a positive proportion of places, and there are twisted derived equivalent K3 surfaces which become derived equivalent upon reduction for a positive proportion of places.

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Section: Introductionmentioning

confidence: 99%

Reduction of Brauer classes on K3 surfaces, rationality and derived equivalence

Frei¹,

Hassett²,

Várilly-Alvarado³

2021

Preprint

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show abstract

“…Moreover, the modularity on orthogonal Shimura varieties in the arithmetic divisor case is proved for maximal quadratic lattices in [15,, which has applications to exceptional jumps of Picard ranks of reductions of K3 surfaces over number fields [49].…”

Section: Introductionmentioning

confidence: 99%

Maximal parahoric arithmetic transfers, resolutions and modularity

Zhang¹

2021

Preprint

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For any unramified quadratic extension of p-adic local fields F/F 0 (p > 2), we formulate several arithmetic transfer conjectures at any maximal parahoric level, in the context of Zhang's relative trace formula approach to the arithmetic Gan-Gross-Prasad conjecture. The singularity is resolved via an arithmetic Atiyah flop using formal Balloon-Ground stratification. Then we do intersections and modify derived fixed points on the resolution.By a local-global method and double induction, we prove these conjectures for F 0 unramified over Qp, including the arithmetic fundamental lemma for p > 2. We introduce the relative Cayley map and also establish Jacquet-Rallis transfers of vertex lattices. We also prove new modularity results for arithmetic theta series at maximal parahoric levels via modifications over Fq and C. Along the way, we study the complex and mod p geometry of Shimura varieties and special cycles through natural stratifications.

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

Cited by 2 publications

References 32 publications

Reduction of Brauer classes on K3 surfaces, rationality and derived equivalence

Reduction of Brauer classes on K3 surfaces, rationality and derived equivalence

Maximal parahoric arithmetic transfers, resolutions and modularity

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