On hyperelliptic Jacobians with complex multiplication by Q(−2+2)

IF 1.2 3区数学 Q1 MATHEMATICS Finite Fields and Their Applications Pub Date : 2024-10-07 DOI:10.1016/j.ffa.2024.102512

Tomasz Jędrzejak

引用次数: 0

Abstract

Consider a one-parameter family of hyperelliptic curves

C_{a} : y^{2} = x^{5} + 3 a x^{4} - 2 a^{2} x^{3} - 6 a^{3} x^{2} + 3 a^{4} x + a^{5}

defined over

Q

, and their Jacobians

J_{a}

where without loss of generality a is a non-zero squarefree integer. Clearly, the curve

C_{a}

is a quadratic twist by a of

C_{1}

. Note that

J_{a}

has complex multiplication by the quartic field

Q (\sqrt{- 2 + \sqrt{2}})

. For a prime

p ∤ 2 a

we obtain types of reduction (supersingular, superspecial, ordinary) of

J_{a}

at p in terms of congruences modulo 16 and the exact formulas for the zeta function of

C_{a}

over

F_{p}

for

p ≢ 1, 7 (\mod 16)

. We deduce as conclusions the complete characterization of torsion subgroups of

J_{a} (Q)

, namely

J_{a} {(Q)}_{tors} = J_{a} (Q) [2] ≃ Z / 2 Z

, and some information about

rank J_{a} (Q)

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关于 Q(-2+2) 复乘法的超椭圆雅各比

考虑定义在 Q 上的超椭圆曲线 Ca:y2=x5+3ax4-2a2x3-6a3x2+3a4x+a5 的单参数族，以及它们的 Jacobian Ja（在不失一般性的前提下，a 为非零的无平方整数）。显然，曲线 Ca 是 C1 的 a 二次扭曲。请注意，Ja 与四元数域 Q(-2+2) 有复乘法关系。对于素数 p∤2a，我们根据同余式 modulo 16 得到了 Ja 在 p 处的还原类型（超同余式、超特异式、普通式），以及对于 p≢1,7(mod16)，Ca 在 Fp 上的 zeta 函数的精确公式。作为结论，我们推导出 Ja(Q) 扭转子群的完整表征，即 Ja(Q)tors=Ja(Q)[2]≃Z/2Z 以及关于秩 Ja(Q) 的一些信息。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Finite Fields and Their Applications 数学-数学

CiteScore

2.00

自引率

20.00%

发文量

133

审稿时长

6-12 weeks

期刊介绍： Finite Fields and Their Applications is a peer-reviewed technical journal publishing papers in finite field theory as well as in applications of finite fields. As a result of applications in a wide variety of areas, finite fields are increasingly important in several areas of mathematics, including linear and abstract algebra, number theory and algebraic geometry, as well as in computer science, statistics, information theory, and engineering. For cohesion, and because so many applications rely on various theoretical properties of finite fields, it is essential that there be a core of high-quality papers on theoretical aspects. In addition, since much of the vitality of the area comes from computational problems, the journal publishes papers on computational aspects of finite fields as well as on algorithms and complexity of finite field-related methods. The journal also publishes papers in various applications including, but not limited to, algebraic coding theory, cryptology, combinatorial design theory, pseudorandom number generation, and linear recurring sequences. There are other areas of application to be included, but the important point is that finite fields play a nontrivial role in the theory, application, or algorithm.

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