Explicit uniform bounds for Brauer groups of singular K3 surfaces

Francesca Balestrieri; Alexis Johnson; Rachel Newton

Explicit uniform bounds for Brauer groups of singular K3 surfaces

Francesca Balestrieri, Alexis Johnson, Rachel Newton

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Abstract

Let k be a number field. We give an explicit bound, depending only on [k: Q], on the size of the Brauer group of a K3 surface X/k that is geometrically isomorphic to the Kummer surface attached to a product of CM elliptic curves. As an application, we show that the Brauer–Manin set for such a variety is effectively computable. In addition, we prove an effective version of the strong Shafarevich conjecture for singular K3 surfaces by giving an explicit bound, depending only on [k: Q], on the number of C-isomorphism classes of singular K3 surfaces defined over k.

MSC Codes 14F22 (Primary) 14J28, 14G05 (Secondary)

Original language	English (US)
Journal	Unknown Journal
State	Published - Jun 26 2020

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abstract = "Let k be a number field. We give an explicit bound, depending only on [k: Q], on the size of the Brauer group of a K3 surface X/k that is geometrically isomorphic to the Kummer surface attached to a product of CM elliptic curves. As an application, we show that the Brauer–Manin set for such a variety is effectively computable. In addition, we prove an effective version of the strong Shafarevich conjecture for singular K3 surfaces by giving an explicit bound, depending only on [k: Q], on the number of C-isomorphism classes of singular K3 surfaces defined over k.MSC Codes 14F22 (Primary) 14J28, 14G05 (Secondary)",

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AU - Johnson, Alexis

AU - Newton, Rachel

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N2 - Let k be a number field. We give an explicit bound, depending only on [k: Q], on the size of the Brauer group of a K3 surface X/k that is geometrically isomorphic to the Kummer surface attached to a product of CM elliptic curves. As an application, we show that the Brauer–Manin set for such a variety is effectively computable. In addition, we prove an effective version of the strong Shafarevich conjecture for singular K3 surfaces by giving an explicit bound, depending only on [k: Q], on the number of C-isomorphism classes of singular K3 surfaces defined over k.MSC Codes 14F22 (Primary) 14J28, 14G05 (Secondary)

AB - Let k be a number field. We give an explicit bound, depending only on [k: Q], on the size of the Brauer group of a K3 surface X/k that is geometrically isomorphic to the Kummer surface attached to a product of CM elliptic curves. As an application, we show that the Brauer–Manin set for such a variety is effectively computable. In addition, we prove an effective version of the strong Shafarevich conjecture for singular K3 surfaces by giving an explicit bound, depending only on [k: Q], on the number of C-isomorphism classes of singular K3 surfaces defined over k.MSC Codes 14F22 (Primary) 14J28, 14G05 (Secondary)

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JO - Unknown Journal

JF - Unknown Journal

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Explicit uniform bounds for Brauer groups of singular K3 surfaces

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