Fully maximal and fully minimal abelian varieties

Valentijn Karemaker; Rachel Pries

doi:10.1016/j.jpaa.2018.10.007

Fully maximal and fully minimal abelian varieties

Valentijn Karemaker^*, Rachel Pries

^*Corresponding author for this work

Fundamental mathematics

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

Abstract

We introduce and study a new way to categorize supersingular abelian varieties defined over a finite field by classifying them as fully maximal, mixed or fully minimal. The type of A depends on the normalized Weil numbers of A and its twists. We analyze these types for supersingular abelian varieties and curves under conditions on the automorphism group. In particular, we present a complete analysis of these properties for supersingular elliptic curves and supersingular abelian surfaces in arbitrary characteristic, and for a one-dimensional family of supersingular curves of genus 3 in characteristic 2.

Original language	English
Pages (from-to)	3031-3056
Number of pages	26
Journal	Journal of Pure and Applied Algebra
Volume	223
Issue number	7
DOIs	https://doi.org/10.1016/j.jpaa.2018.10.007
Publication status	Published - Jul 2019

Bibliographical note

Funding Information:
Karemaker was partially supported by The Netherlands Organisation for Scientific Research (NWO) through the “Geometry and Quantum Theory” research cluster. Pries was partially supported by NSF grant DMS-15-02227. The authors thank Jeff Achter, Gunther Cornelissen, Frans Oort, Christophe Ritzenthaler, Jeroen Sijsling, Andrew Sutherland, and some referees for helpful comments.

Funding Information:
Karemaker was partially supported by The Netherlands Organisation for Scientific Research (NWO) through the ?Geometry and Quantum Theory? research cluster. Pries was partially supported by NSF grant DMS-15-02227. The authors thank Jeff Achter, Gunther Cornelissen, Frans Oort, Christophe Ritzenthaler, Jeroen Sijsling, Andrew Sutherland, and some referees for helpful comments.

Publisher Copyright:
© 2018 Elsevier B.V.

Funding

Karemaker was partially supported by The Netherlands Organisation for Scientific Research (NWO) through the “Geometry and Quantum Theory” research cluster. Pries was partially supported by NSF grant DMS-15-02227. The authors thank Jeff Achter, Gunther Cornelissen, Frans Oort, Christophe Ritzenthaler, Jeroen Sijsling, Andrew Sutherland, and some referees for helpful comments. Karemaker was partially supported by The Netherlands Organisation for Scientific Research (NWO) through the ?Geometry and Quantum Theory? research cluster. Pries was partially supported by NSF grant DMS-15-02227. The authors thank Jeff Achter, Gunther Cornelissen, Frans Oort, Christophe Ritzenthaler, Jeroen Sijsling, Andrew Sutherland, and some referees for helpful comments.

Keywords

Abelian variety
Curve
Maximal
Supersingular
Weil number
Zeta function

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10.1016/j.jpaa.2018.10.007

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abstract = "We introduce and study a new way to categorize supersingular abelian varieties defined over a finite field by classifying them as fully maximal, mixed or fully minimal. The type of A depends on the normalized Weil numbers of A and its twists. We analyze these types for supersingular abelian varieties and curves under conditions on the automorphism group. In particular, we present a complete analysis of these properties for supersingular elliptic curves and supersingular abelian surfaces in arbitrary characteristic, and for a one-dimensional family of supersingular curves of genus 3 in characteristic 2.",

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note = "Funding Information: Karemaker was partially supported by The Netherlands Organisation for Scientific Research (NWO) through the “Geometry and Quantum Theory” research cluster. Pries was partially supported by NSF grant DMS-15-02227. The authors thank Jeff Achter, Gunther Cornelissen, Frans Oort, Christophe Ritzenthaler, Jeroen Sijsling, Andrew Sutherland, and some referees for helpful comments. Funding Information: Karemaker was partially supported by The Netherlands Organisation for Scientific Research (NWO) through the ?Geometry and Quantum Theory? research cluster. Pries was partially supported by NSF grant DMS-15-02227. The authors thank Jeff Achter, Gunther Cornelissen, Frans Oort, Christophe Ritzenthaler, Jeroen Sijsling, Andrew Sutherland, and some referees for helpful comments. Publisher Copyright: {\textcopyright} 2018 Elsevier B.V.",

year = "2019",

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doi = "10.1016/j.jpaa.2018.10.007",

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AU - Pries, Rachel

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AB - We introduce and study a new way to categorize supersingular abelian varieties defined over a finite field by classifying them as fully maximal, mixed or fully minimal. The type of A depends on the normalized Weil numbers of A and its twists. We analyze these types for supersingular abelian varieties and curves under conditions on the automorphism group. In particular, we present a complete analysis of these properties for supersingular elliptic curves and supersingular abelian surfaces in arbitrary characteristic, and for a one-dimensional family of supersingular curves of genus 3 in characteristic 2.

KW - Abelian variety

KW - Curve

KW - Maximal

KW - Supersingular

KW - Weil number

KW - Zeta function

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M3 - Article

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SP - 3031

EP - 3056

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

IS - 7

ER -

Fully maximal and fully minimal abelian varieties

Abstract

Bibliographical note

Funding

Keywords

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