The Tate conjecture for K3 surfaces in odd characteristic

Pera, Keerthi Madapusi

doi:10.1007/s00222-014-0557-5

Cited by 106 publications

(60 citation statements)

References 33 publications

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“…For non-supersingular K3 surfaces in characteristic p > 5, the Tate conjecture was proved by Nygaard and Ogus in 1985 [34]. Finally, the supersingular case was proved in a series of papers starting in 2012 by Charles, Kim, Madapusi Pera, and Maulik [7,20,26,27]. The case of characteristic 2 was proved only in the last of these papers, by Kim and Madapusi Pera.…”

Section: Further Results Over Finite Fieldsmentioning

confidence: 99%

Recent progress on the Tate conjecture

Totaro

2017

Bull. Amer. Math. Soc.

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Section: Further Results Over Finite Fieldsmentioning

confidence: 99%

Recent progress on the Tate conjecture

Totaro

2017

Bull. Amer. Math. Soc.

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“…[DP94,KR99]. Moreover, there is work in progress by Kisin and Pappas, which will generalize the methods of [Kis10] to construct integral models of Shimura varieties of abelian type with general parahoric level at p. However, the simple and direct nature of our construction seems to be quite useful in applications, which include the Tate conjecture for K3 surfaces in odd characteristic [MaP14], and also forthcoming work, in collaboration with F. Andreatta, E. Goren and B. Howard, on the arithmetic intersection theory over these Shimura varieties.…”

Section: Introductionmentioning

confidence: 99%

Integral canonical models for Spin Shimura varieties

Pera¹

2015

Compositio Math.

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109

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We construct regular integral canonical models for Shimura varieties attached to Spin groups at (possibly ramified) odd primes. We exhibit these models as schemes of 'relative PEL type' over integral canonical models of larger Spin Shimura varieties with good reduction. Work of Vasiu-Zink then shows that the classical Kuga-Satake construction extends over the integral model and that the integral models we construct are canonical in a very precise sense. We also construct good compactifications for our integral models. Our results have applications to the Tate conjecture for K3 surfaces, as well as to Kudla's program of relating intersection numbers of special cycles on orthogonal Shimura varieties to Fourier coefficients of modular forms

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“…As mentioned in §2.2, this has been confirmed for all known examples except OG6. [48] has dealt with the case of K3 surfaces in adelic language (see also [38,33]), but there is no difficulty to extend his construction to hyperkähler manifolds, see e.g. [16, §3.3].…”

Section: Denote Bymentioning

confidence: 99%

Tautological classes on moduli spaces of hyper-Kähler manifolds

Bergeron¹,

Li²

2019

Duke Math. J.

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In this paper, we discuss the cycle theory on moduli spaces F h of h-polarized hyperkähler manifolds. Firstly, we construct the tautological ring on F h following the work of Marian, Oprea and Pandharipande on the tautological conjecture on moduli spaces of K3 surfaces. We study the tautological classes in cohomology groups and prove that most of them are linear combinations of Noether-Lefschetz cycle classes. In particular, we prove the cohomological version of the tautological conjecture on moduli space of K3 [n] -type hyperkähler manifolds with n ≤ 2. Secondly, we prove the cohomological generalized Franchetta conjecture on universal family of these hyperkähler manifolds.

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The Tate conjecture for K3 surfaces in odd characteristic

Cited by 106 publications

References 33 publications

Recent progress on the Tate conjecture

Recent progress on the Tate conjecture

Integral canonical models for Spin Shimura varieties

Tautological classes on moduli spaces of hyper-Kähler manifolds

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