Maximal curves and Tate-Shafarevich results for quartic and sextic twists

Sven Bootsma; Saeed Tafazolian; Jaap Top

doi:10.1016/j.ffa.2023.102256

Maximal curves and Tate-Shafarevich results for quartic and sextic twists

Sven Bootsma, Saeed Tafazolian, Jaap Top^*

^*Corresponding author for this work

Algebra

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Abstract

We study elliptic surfaces corresponding to an equation of the specific type y²=x³+f(t)x, defined over the finite field F_q for a prime power q≡3mod4. It is shown that if s⁴=f(t) defines a curve that is maximal over F_q² then the rank of the group of sections defined over F_q on the elliptic surface is determined in terms of elementary properties of the rational function f(t). Similar results are shown for elliptic surfaces given by y²=x³+g(t) using prime powers q≡5mod6 and curves s⁶=g(t). Finally, for each of the forms used here, existence of curves with the property that they are maximal over F_q² is discussed, as well as various examples.

Original language	English
Article number	102256
Number of pages	13
Journal	Finite fields and their applications
Volume	91
DOIs	https://doi.org/10.1016/j.ffa.2023.102256
Publication status	Published - Oct-2023

Keywords

Elliptic curve
Elliptic surface
Finite field
Function field
Maximal curve
Mordell-Weil rank

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Maximal curves and Tate-Shafarevich results for quartic and sextic twistsFinal publisher's version, 354 KBLicence: CC BY

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@article{1651e354f9aa4f699861a0bbd95bfbce,

title = "Maximal curves and Tate-Shafarevich results for quartic and sextic twists",

abstract = "We study elliptic surfaces corresponding to an equation of the specific type y2=x3+f(t)x, defined over the finite field Fq for a prime power q≡3mod4. It is shown that if s4=f(t) defines a curve that is maximal over Fq2 then the rank of the group of sections defined over Fq on the elliptic surface is determined in terms of elementary properties of the rational function f(t). Similar results are shown for elliptic surfaces given by y2=x3+g(t) using prime powers q≡5mod6 and curves s6=g(t). Finally, for each of the forms used here, existence of curves with the property that they are maximal over Fq2 is discussed, as well as various examples.",

keywords = "Elliptic curve, Elliptic surface, Finite field, Function field, Maximal curve, Mordell-Weil rank",

author = "Sven Bootsma and Saeed Tafazolian and Jaap Top",

note = "Funding Information: The second author was partially supported by FAPESP grant No. 2022/06589-5 and by CNPq grant No. 310194/2019-9. Publisher Copyright: {\textcopyright} 2023 The Author(s)",

year = "2023",

month = oct,

doi = "10.1016/j.ffa.2023.102256",

language = "English",

volume = "91",

journal = "Finite fields and their applications",

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TY - JOUR

T1 - Maximal curves and Tate-Shafarevich results for quartic and sextic twists

AU - Bootsma, Sven

AU - Tafazolian, Saeed

AU - Top, Jaap

PY - 2023/10

Y1 - 2023/10

N2 - We study elliptic surfaces corresponding to an equation of the specific type y2=x3+f(t)x, defined over the finite field Fq for a prime power q≡3mod4. It is shown that if s4=f(t) defines a curve that is maximal over Fq2 then the rank of the group of sections defined over Fq on the elliptic surface is determined in terms of elementary properties of the rational function f(t). Similar results are shown for elliptic surfaces given by y2=x3+g(t) using prime powers q≡5mod6 and curves s6=g(t). Finally, for each of the forms used here, existence of curves with the property that they are maximal over Fq2 is discussed, as well as various examples.

AB - We study elliptic surfaces corresponding to an equation of the specific type y2=x3+f(t)x, defined over the finite field Fq for a prime power q≡3mod4. It is shown that if s4=f(t) defines a curve that is maximal over Fq2 then the rank of the group of sections defined over Fq on the elliptic surface is determined in terms of elementary properties of the rational function f(t). Similar results are shown for elliptic surfaces given by y2=x3+g(t) using prime powers q≡5mod6 and curves s6=g(t). Finally, for each of the forms used here, existence of curves with the property that they are maximal over Fq2 is discussed, as well as various examples.

KW - Elliptic curve

KW - Elliptic surface

KW - Finite field

KW - Function field

KW - Maximal curve

KW - Mordell-Weil rank

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DO - 10.1016/j.ffa.2023.102256

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SN - 1071-5797

VL - 91

JO - Finite fields and their applications

JF - Finite fields and their applications

M1 - 102256

ER -

Maximal curves and Tate-Shafarevich results for quartic and sextic twists

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Keywords

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