On the maximality of hyperelliptic Howe curves of genus 3

Ohashi, Ryo

doi:10.2996/kmj45206

Cited by 2 publications

(1 citation statement)

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“…On the other hand, it follows from Auer-Top's result [1,Proposition 2.2] that all supersingular Legendre elliptic curves E : y 2 = x(x − 1)(x − t) are maximal or minimal over F p 2 . In [16,Theorem 1.1], the author studied the maximality of hyperelliptic genus-3 curves H : y 2 = (x 4 − ax 2 + 1)(x 4 − bx 2 + 1), whose automorphism groups contain (Z/2Z) 3 ; if H is superspecial, then H is maximal or minimal over F p 2 . In [17,Theorem 1.1], the author also studied the maximality of Ciani quartics H ′ : x 4 + y 4 + z 4 + rx 2 y 2 + sy 2 z 2 + tz 2 x 2 = 0, which are non-hyperelliptic genus-3 curves whose automorphism groups contain the dihedral group of order 4; if H ′ is superspecial, then H ′ is maximal or minimal over F p 2 .…”

Section: Introductionmentioning

confidence: 99%

Fast Enumeration of Superspecial Hyperelliptic Curves of Genus 4 with Automorphism Group $$V_4$$

Ohashi

Kudo²,

Harashita³

2023

Lecture Notes in Computer Science

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In this paper, we examine superspecial genus-2 curves C :As a main result, we show that the difference between any two elements in {0, 1, λ, µ, ν} is a square in F p 2 . Moreover, we show that C is maximal or minimal over F p 2 without taking its F p 2 -form (we also give a criterion in terms of p that tells whether C is maximal or minimal). As these applications, we study the maximality of superspecial hyperelliptic curves of genus 3 and 4 whose automorphism groups contain Z/2Z × Z/2Z. Rosenhain forms of genus-2 curveAny genus-2 curve has just 6 Weierstrass points, and we can consider an isomorphism C ∼ = C λ,µ,ν which maps three of them to 0, 1 and ∞.Definition 2.1. Given a genus-2 curve C, we say thatis a Rosenhain form of C when there exists an isomorphism C ∼ = C λ,µ,ν over the algebraic closure of K. Then the values λ, µ and ν are called Rosenhain invariants of C λ,µ,ν .

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