Families of supersingular curves in characteristic 2

Jasper Scholten; Hui June Zhu

doi:10.4310/MRL.2002.v9.n5.a7

Families of supersingular curves in characteristic 2

Jasper Scholten
, Hui June Zhu

Mathematics

KU Leuven

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

10 Scopus citations

Abstract

This paper determines normal forms of all hyperelliptic supersingular curves of genus g over an algebraically closed field F of characteristic 2 for 1 ≤ g ≤ 8. We also show that every hyperelliptic supersingular curve of genus 9 over F has an equation y² - y = x¹⁹ + c⁸x⁹ + c³x for some c ∈ F̄₂. Consequently, the paper determines the dimensions of the open locus of hyperelliptic supersingular curves of genus g ≤ 9 over F̄₂.

Original language	English
Pages (from-to)	639-650
Number of pages	12
Journal	Mathematical Research Letters
Volume	9
Issue number	5-6
DOIs	https://doi.org/10.4310/MRL.2002.v9.n5.a7
State	Published - 2002

Keywords

Hyperelhptic curves
Moduli space of curves
Normal forms
Supersingular curves

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10.4310/MRL.2002.v9.n5.a7

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title = "Families of supersingular curves in characteristic 2",

abstract = "This paper determines normal forms of all hyperelliptic supersingular curves of genus g over an algebraically closed field F of characteristic 2 for 1 ≤ g ≤ 8. We also show that every hyperelliptic supersingular curve of genus 9 over F has an equation y2 - y = x19 + c8x9 + c3x for some c ∈ {\=F}2. Consequently, the paper determines the dimensions of the open locus of hyperelliptic supersingular curves of genus g ≤ 9 over {\=F}2.",

keywords = "Hyperelhptic curves, Moduli space of curves, Normal forms, Supersingular curves",

author = "Jasper Scholten and Zhu, \{Hui June\}",

year = "2002",

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T1 - Families of supersingular curves in characteristic 2

AU - Scholten, Jasper

AU - Zhu, Hui June

PY - 2002

Y1 - 2002

N2 - This paper determines normal forms of all hyperelliptic supersingular curves of genus g over an algebraically closed field F of characteristic 2 for 1 ≤ g ≤ 8. We also show that every hyperelliptic supersingular curve of genus 9 over F has an equation y2 - y = x19 + c8x9 + c3x for some c ∈ F̄2. Consequently, the paper determines the dimensions of the open locus of hyperelliptic supersingular curves of genus g ≤ 9 over F̄2.

AB - This paper determines normal forms of all hyperelliptic supersingular curves of genus g over an algebraically closed field F of characteristic 2 for 1 ≤ g ≤ 8. We also show that every hyperelliptic supersingular curve of genus 9 over F has an equation y2 - y = x19 + c8x9 + c3x for some c ∈ F̄2. Consequently, the paper determines the dimensions of the open locus of hyperelliptic supersingular curves of genus g ≤ 9 over F̄2.

KW - Hyperelhptic curves

KW - Moduli space of curves

KW - Normal forms

KW - Supersingular curves

UR - https://www.scopus.com/pages/publications/0036771387

U2 - 10.4310/MRL.2002.v9.n5.a7

DO - 10.4310/MRL.2002.v9.n5.a7

M3 - Article

AN - SCOPUS:0036771387

SN - 1073-2780

VL - 9

SP - 639

EP - 650

JO - Mathematical Research Letters

JF - Mathematical Research Letters

IS - 5-6

ER -

Families of supersingular curves in characteristic 2

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Keywords

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