EVERY BT1 GROUP SCHEME APPEARS in A JACOBIAN

Rachel Pries; Douglas Ulmer

doi:10.1090/proc/15681

EVERY BT₁ GROUP SCHEME APPEARS in A JACOBIAN

Rachel Pries, Douglas Ulmer

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

1 Scopus citations

Abstract

Let p be a prime number and let k be an algebraically closed field of characteristic p. A BT₁ group scheme over k is a finite commutative group scheme which arises as the kernel of p on a p-divisible (Barsotti–Tate) group. Our main result is that every BT₁ group scheme over k occurs as a direct factor of the p-torsion group scheme of the Jacobian of an explicit curve defined over F_p. We also treat a variant with polarizations. Our main tools are the Kraft classification of BT₁ group schemes, a theorem of Oda, and a combinatorial description of the de Rham cohomology of Fermat curves.

Original language	English (US)
Pages (from-to)	525-537
Number of pages	13
Journal	Proceedings of the American Mathematical Society
Volume	150
Issue number	2
DOIs	https://doi.org/10.1090/proc/15681
State	Published - 2022
Externally published	Yes

Keywords

Abelian variety
Curve
De Rham cohomology
Dieudonné module
Fermat curve
Finite field
Frobenius
Group scheme
Jacobian
P-divisible group
Verschiebung

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

Access to Document

10.1090/proc/15681

Cite this

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title = "EVERY BT1 GROUP SCHEME APPEARS in A JACOBIAN",

abstract = "Let p be a prime number and let k be an algebraically closed field of characteristic p. A BT1 group scheme over k is a finite commutative group scheme which arises as the kernel of p on a p-divisible (Barsotti–Tate) group. Our main result is that every BT1 group scheme over k occurs as a direct factor of the p-torsion group scheme of the Jacobian of an explicit curve defined over Fp. We also treat a variant with polarizations. Our main tools are the Kraft classification of BT1 group schemes, a theorem of Oda, and a combinatorial description of the de Rham cohomology of Fermat curves.",

keywords = "Abelian variety, Curve, De Rham cohomology, Dieudonn{\'e} module, Fermat curve, Finite field, Frobenius, Group scheme, Jacobian, P-divisible group, Verschiebung",

author = "Rachel Pries and Douglas Ulmer",

note = "Publisher Copyright: 2021 American Mathematical Society",

year = "2022",

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language = "English (US)",

volume = "150",

pages = "525--537",

journal = "Proceedings of the American Mathematical Society",

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T1 - EVERY BT1 GROUP SCHEME APPEARS in A JACOBIAN

AU - Pries, Rachel

AU - Ulmer, Douglas

N1 - Publisher Copyright: 2021 American Mathematical Society

PY - 2022

Y1 - 2022

N2 - Let p be a prime number and let k be an algebraically closed field of characteristic p. A BT1 group scheme over k is a finite commutative group scheme which arises as the kernel of p on a p-divisible (Barsotti–Tate) group. Our main result is that every BT1 group scheme over k occurs as a direct factor of the p-torsion group scheme of the Jacobian of an explicit curve defined over Fp. We also treat a variant with polarizations. Our main tools are the Kraft classification of BT1 group schemes, a theorem of Oda, and a combinatorial description of the de Rham cohomology of Fermat curves.

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KW - Curve

KW - De Rham cohomology

KW - Dieudonné module

KW - Fermat curve

KW - Finite field

KW - Frobenius

KW - Group scheme

KW - Jacobian

KW - P-divisible group

KW - Verschiebung

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