Finite Fields and Their Applications最新文献

英文中文

Stable binomials over finite fields 有限域上的稳定二项式

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-10-18 DOI: 10.1016/j.ffa.2024.102520

Arthur Fernandes , Daniel Panario , Lucas Reis

In this paper, we study stable binomials over finite fields, i.e., irreducible binomials

x^{t} - b \in F_{q} [x]

such that all their iterates are also irreducible over

F_{q}

. We obtain a simple criterion on the stability of binomials based on the forward orbit of 0 under the map

z \mapsto z^{t} - b

. In particular, our criterion extends the one obtained by Jones and Boston (2011) for the quadratic case. As applications of our main result, we obtain an explicit 1-parameter family of stable quartics over prime fields

F_{p}

with

p \equiv 5 (\mod 24)

and also develop an algorithm to test the stability of binomials over finite fields. Finally, building upon a work of Ostafe and Shparlinski (2010), we employ character sums to bound the complexity of such algorithm.

本文研究有限域上的稳定二项式，即不可约二项式 xt-b∈Fq[x]，使得它们的所有迭代也都是 Fq 上的不可约二项式。我们根据 0 在 z↦zt-b 映射下的前向轨道，得到了一个关于二项式稳定性的简单判据。特别是，我们的判据扩展了 Jones 和 Boston（2011）在二次情况下得到的判据。作为我们主要结果的应用，我们得到了质域 Fp 上 p≡5(mod24)的稳定四元数的一个明确的 1 参数族，还开发了一种算法来检验有限域上二项式的稳定性。最后，在 Ostafe 和 Shparlinski（2010 年）工作的基础上，我们利用特征和来约束这种算法的复杂性。

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引用次数: 0

Cyclic locally recoverable LCD codes with the help of cyclotomic polynomials 借助循环多项式的循环局部可恢复液晶编码

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-10-15 DOI: 10.1016/j.ffa.2024.102519

Anuj Kumar Bhagat, Ritumoni Sarma

This article explores two families of cyclic codes over

F_{q}

of length n denoted by

C_{n}

and

C_{n, 1}

, which are generated by the n-th cyclotomic polynomial

Q_{n} (x)

and the polynomial

Q_{n} (x) Q_{1} (x)

, respectively. We find formulae for the distance of

C_{n}

and

C_{n, 1}

for each

n > 1

and conjecture formulae for the distance of their (Euclidean) duals. We prove the conjecture when n is a product of at most two distinct prime powers. Moreover, we show that all these codes are LCD codes, and several subfamilies are both r-optimal and d-optimal locally recoverable codes.

本文探讨了长度为 n 的 Fq 上的两个循环码族，分别用 Cn 和 Cn,1 表示，它们分别由 n 次循环多项式 Qn(x) 和多项式 Qn(x)Q1(x) 生成。我们找到了每个 n>1 的 Cn 和 Cn,1 的距离公式，并猜想了它们（欧几里得）对偶的距离公式。当 n 是最多两个不同质幂的乘积时，我们证明了猜想。此外，我们还证明了所有这些编码都是 LCD 编码，而且有几个子系列既是 r-最优编码，也是 d-最优局部可恢复编码。

引用次数: 0

Some new constructions of optimal and almost optimal locally repairable codes 最优和近似最优局部可修复代码的一些新构造

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-10-15 DOI: 10.1016/j.ffa.2024.102518

Varsha Chauhan, Anuradha Sharma

<div><div>Additive codes over finite fields are natural extensions of linear codes and are useful in constructing quantum error-correcting codes. In this paper, we first study the locality properties of additive MDS codes over finite fields whose dual codes are also MDS. We further provide a method to construct optimal and almost optimal LRCs with new parameters belonging to the family of additive codes, which are not MDS. More precisely, for an integer <math><msub><mrow><mi>m</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>≥</mo><mn>2</mn></math> and a prime power q, we provide a method to construct optimal and almost optimal <math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math>-additive LRCs over <math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><msub><mrow><mi>m</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></mrow></msub></math> with locality r that relies on the existence of certain special polynomials over <math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math>, which we shall refer to as <math><mo>(</mo><mi>r</mi><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></math>-good polynomials over <math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math>, (note that <math><mo>(</mo><mi>r</mi><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></math>-good polynomials over <math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math> coincide with r-good polynomials over <math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math> when <math><msub><mrow><mi>m</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mn>1</mn></math>). We also derive sufficient conditions under which <math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math>-additive LRCs over <math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><msub><mrow><mi>m</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></mrow></msub></math> constructed using the aforementioned method are optimal. We further provide four general methods to construct <math><mo>(</mo><mi>r</mi><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></math>-good polynomials over <math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math>, which give rise to several classes of optimal and almost optimal LRCs over <math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><msub><mrow><mi>m</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></mrow></msub></math> with locality r. To illustrate these results, we list several optimal LRCs over <math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><msub><mrow><m

有限域上的加法码是线性码的自然扩展，在构建量子纠错码时非常有用。在本文中，我们首先研究了有限域上加法 MDS 码的局部性特性，其对偶码也是 MDS。我们进一步提供了一种方法，用属于非 MDS 的加法码系列的新参数来构造最优和几乎最优的 LRC。更确切地说，对于整数 m0≥2 和素数幂 q，我们提供了一种方法来构造 Fqm0 上具有局部性 r 的最优和近似最优 Fq 附加 LRC，这种方法依赖于 Fq 上某些特殊多项式的存在，我们将其称为 Fq 上的（r,m0）-好多项式（注意，当 m0=1 时，Fq 上的（r,m0）-好多项式与 Fq 上的（r-好多项式）重合）。我们还推导出充分条件，在这些条件下，用上述方法构造的 Fq 上的 Fq-additive LRC 是最优的。为了说明这些结果，我们列出了几个带有新参数的 Fqm0 上最优 LRC。最后，我们考虑了 m0=1 的情况，得到了 Fq 上一些新的 r-good 多项式，从而构建了具有局部性 r 的 Fq 上最优线性 LRC。

引用次数: 0

The generalized Suzuki curve 广义铃木曲线

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-10-11 DOI: 10.1016/j.ffa.2024.102514

Saeed Tafazolian

We present a characterization of the generalized Suzuki curve, focusing on its genus and automorphism group.

我们介绍了广义铃木曲线的特征，重点是它的属和自变群。

引用次数: 0

Few-weight linear codes over Fp from t-to-one mappings 从 t 到一映射的 Fp 上的少权线性编码

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-10-11 DOI: 10.1016/j.ffa.2024.102510

René Rodríguez-Aldama

<div><div>For any prime number p, we provide two classes of linear codes with few weights over a p-ary alphabet. These codes are based on a well-known generic construction (the defining-set method), stemming on a class of monomials and a class of trinomials over finite fields. The considered monomials are Dembowski-Ostrom monomials <math><msup><mrow><mi>x</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>α</mi></mrow></msup><mo>+</mo><mn>1</mn></mrow></msup></math>, for a suitable choice of the exponent α, so that, when <math><mi>p</mi><mo>></mo><mn>2</mn></math> and <math><mi>n</mi><mo>≢</mo><mn>0</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>4</mn><mo>)</mo></math>, these monomials are planar. We study the properties of such monomials in detail for each integer <math><mi>n</mi><mo>></mo><mn>1</mn></math> and any prime number p. In particular, we show that they are t-to-one, where the parameter t depends on the field <math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math> and it takes the values <math><mn>1</mn><mo>,</mo><mn>2</mn></math> or <math><mi>p</mi><mo>+</mo><mn>1</mn></math>. Moreover, we give a simple proof of the fact that the functions are δ-uniform with <math><mi>δ</mi><mo>∈</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>,</mo><mi>p</mi><mo>}</mo></math>. This result describes the differential behavior of these monomials for any p and n. For the second class of functions, we consider an affine equivalent trinomial to <math><msup><mrow><mi>x</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>α</mi></mrow></msup><mo>+</mo><mn>1</mn></mrow></msup></math>, namely, <math><msup><mrow><mi>x</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>α</mi></mrow></msup><mo>+</mo><mn>1</mn></mrow></msup><mo>+</mo><mi>λ</mi><msup><mrow><mi>x</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></msup><mo>+</mo><msup><mrow><mi>λ</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></msup><mi>x</mi></math> for <math><mi>λ</mi><mo>∈</mo><msubsup><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow><mrow><mo>⁎</mo></mrow></msubsup></math>. We prove that these trinomials satisfy certain regularity properties, which are useful for the specification of linear codes with three or four weights that are different than the monomial construction. These families of codes contain projective codes and optimal codes (with respect to the Griesmer bound). Remarkably, they contain infinite families of self-orthogonal and minimal p-ary linear codes for every prime number p. Our findings highlight the utility of st

对于任意素数 p，我们提供了两类 pary 字母表上权重较小的线性编码。这些编码基于一种著名的通用构造（定义集方法），源于有限域上的一类单项式和一类三项式。所考虑的单项式是登鲍斯基-奥斯特罗姆单项式 xpα+1，对于指数 α 的适当选择，当 p>2 和 n≢0(mod4)时，这些单项式是平面的。我们详细研究了每个整数 n>1 和任意素数 p 的此类单项式的性质。我们特别证明了它们是 t 对一的，其中参数 t 取决于场 Fpn，取值为 1、2 或 p+1。此外，我们还给出了函数δ∈{1,4,p}的δ均匀性的简单证明。对于第二类函数，我们考虑 xpα+1 的仿射等价三项式，即对于 λ∈Fpn⁎ 的 xpα+1+λxpα+λpαx 。我们证明了这些三项式满足某些正则特性，这对于规范与单项式构造不同的三重或四重线性编码非常有用。这些代码族包含投影代码和最优代码（关于格里斯梅尔约束）。值得注意的是，对于每个素数 p，它们都包含自正交和最小 pary 线性编码的无穷族。我们的发现突出了研究仿射等价函数的实用性，而这一点在这方面往往被忽视。

引用次数: 0

Drinfeld module and Weil pairing over Dedekind domain of class number two 二类 Dedekind 域上的 Drinfeld 模块和 Weil 配对

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-10-10 DOI: 10.1016/j.ffa.2024.102516

Chuangqiang Hu , Xiao-Min Huang

The primary objective of this paper is to derive explicit formulas for rank one and rank two Drinfeld modules over a specific domain denoted by

A

. This domain corresponds to the projective line associated with an infinite place of degree two. To achieve the goals, we construct a pair of standard rank one Drinfeld modules whose coefficients are in the Hilbert class field of

A

. We demonstrate that the period lattice of the exponential functions corresponding to both modules behaves similarly to the period lattice of the Carlitz module, the standard rank one Drinfeld module defined over rational function fields. Moreover, we employ Anderson's t-motive to obtain the complete family of rank two Drinfeld modules. This family is parameterized by the invariant

J = λ^{q^{2} + 1}

which effectively serves as the counterpart of the j-invariant for elliptic curves. Building upon the concepts introduced by van der Heiden, particularly with regard to rank two Drinfeld modules, we are able to reformulate the Weil pairing of Drinfeld modules of any rank using a specialized polynomial in multiple variables known as the Weil operator. As an illustrative example, we provide a detailed examination of a more explicit formula for the Weil pairing and the Weil operator of rank two Drinfeld modules over the domain

A

本文的主要目的是推导出一个特定域（用 A 表示）上的秩一和秩二 Drinfeld 模块的明确公式。为了实现这些目标，我们构建了一对系数在 A 的希尔伯特类域中的标准一级德林菲尔德模块。我们证明，这两个模块对应的指数函数的周期网格与卡利茨模块（定义在有理函数域上的标准一级德林菲尔德模块）的周期网格表现类似。此外，我们利用安德森 t 动力得到了完整的二阶德林菲尔德模块族。这个族的参数是不变式 J=λq2+1，它实际上是椭圆曲线 j 不变式的对应变量。基于范德尔海登提出的概念，特别是关于秩二的 Drinfeld 模块的概念，我们能够使用称为 Weil 算子的专门多变量多项式来重新表述任意秩的 Drinfeld 模块的 Weil 配对。举例说明，我们将详细研究域 A 上的二阶 Drinfeld 模块的 Weil 配对和 Weil 算子的更明确公式。

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引用次数: 0

A spherical extension theorem and applications in positive characteristic 正特征球面扩展定理及其应用

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-10-10 DOI: 10.1016/j.ffa.2024.102515

Doowon Koh , Thang Pham

In this paper, we prove an extension theorem for spheres of square radii in

F_{q}^{d}

, which improves a result obtained by Iosevich and Koh [9] (2010). Our main tool is a new point-hyperplane incidence bound which will be derived via a cone restriction theorem due to the authors and Lee (2022). Applications on the distance problems will be also discussed.

在本文中，我们证明了 Fqd 中方形半径球面的扩展定理，它改进了 Iosevich 和 Koh [9] (2010) 所获得的结果。我们的主要工具是一个新的点-超平面入射界限，它将通过作者和 Lee (2022) 提出的圆锥限制定理得到。我们还将讨论它在距离问题上的应用。

引用次数: 0

Quadratic residue patterns, algebraic curves and a K3 surface 二次残差模式、代数曲线和 K3 曲面

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-10-10 DOI: 10.1016/j.ffa.2024.102517

Valentina Kiritchenko , Michael Tsfasman , Serge Vlăduţ , Ilya Zakharevich

Quadratic residue patterns modulo a prime are studied since 19th century. In the first part we extend existing results on the number of consecutive ℓ-tuples of quadratic residues, studying corresponding algebraic curves and their Jacobians, which happen to be products of Jacobians of hyperelliptic curves. In the second part we state the last unpublished result of Lydia Goncharova on squares such that their differences are also squares, reformulate it in terms of algebraic geometry of a K3 surface, and prove it. The core of this theorem is an unexpected relation between the number of points on the K3 surface and that on a CM elliptic curve.

自 19 世纪以来，人们一直在研究调制素数的二次残差模式。在第一部分中，我们扩展了关于二次残差的连续 ℓ-tuples 数的现有结果，研究了相应的代数曲线及其雅各比，它们恰好是超椭圆曲线雅各比的乘积。在第二部分中，我们陈述了莉迪亚-冈察洛娃（Lydia Goncharova）最后一个未发表的关于正方形的结果，即它们的差也是正方形，用 K3 曲面的代数几何重新表述并证明了这一结果。该定理的核心是 K3 曲面上的点数与 CM 椭圆曲线上的点数之间的意外关系。

引用次数: 0

An approach to the moments subset sum problem through systems of diagonal equations over finite fields 通过有限域对角方程组解决矩子集和问题的方法

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-10-07 DOI: 10.1016/j.ffa.2024.102511

Juan Francisco Gottig , Mariana Pérez , Melina Privitelli

Let

F_{q}

be the finite field of q elements. Let m and k be positive integers,

b \in F_{q}^{m}

and

D \subset F_{q}

with

k \leq | D |

. Our aim is to determine the existence of a subset

S \subset D

of cardinality k such that

\sum_{a \in S} a^{i} = b_{i}

for

i = 1, \dots, m

. This problem is known as the moments subset sum problem. We take a novel approach to this problem by establishing a connection between the existence of such a subset S with the problem of determining if a certain system of diagonal equations has at least one rational solution with distinct coordinates. To achieve this, we study some relevant geometric properties of the set of solutions of the mentioned system. This analysis allows us to provide estimates on the number of rational solutions of systems of diagonal equations and we subsequently apply these results to address the moments subset sum problem.

设 Fq 是有 q 个元素的有限域。设 m 和 k 为正整数，b∈Fqm，D⊂Fq，k≤|D|。我们的目标是确定是否存在一个心数为 k 的子集 S⊂D，使得 i=1,...m 时，∑a∈Sai=bi。这个问题被称为矩子集和问题。我们对这一问题采取了一种新的方法，即在这样一个子集 S 的存在与确定某个对角方程组是否至少有一个具有不同坐标的有理解这一问题之间建立联系。为此，我们研究了上述系统解集的一些相关几何性质。通过分析，我们可以对对角方程组的有理解的数量进行估计，随后我们将应用这些结果来解决矩子集和问题。

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引用次数: 0

On hyperelliptic Jacobians with complex multiplication by Q(−2+2) 关于 Q(-2+2) 复乘法的超椭圆雅各比

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

<div><div>Consider a one-parameter family of hyperelliptic curves <math><msub><mrow><mi>C</mi></mrow><mrow><mi>a</mi></mrow></msub><mo>:</mo><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>5</mn></mrow></msup><mo>+</mo><mn>3</mn><mi>a</mi><msup><mrow><mi>x</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>−</mo><mn>2</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>−</mo><mn>6</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>3</mn></mrow></msup><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>3</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>4</mn></mrow></msup><mi>x</mi><mo>+</mo><msup><mrow><mi>a</mi></mrow><mrow><mn>5</mn></mrow></msup></math> defined over <math><mi>Q</mi></math>, and their Jacobians <math><msub><mrow><mi>J</mi></mrow><mrow><mi>a</mi></mrow></msub></math> where without loss of generality a is a non-zero squarefree integer. Clearly, the curve <math><msub><mrow><mi>C</mi></mrow><mrow><mi>a</mi></mrow></msub></math> is a quadratic twist by a of <math><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub></math>. Note that <math><msub><mrow><mi>J</mi></mrow><mrow><mi>a</mi></mrow></msub></math> has complex multiplication by the quartic field <math><mi>Q</mi><mrow><mo>(</mo><msqrt><mrow><mo>−</mo><mn>2</mn><mo>+</mo><msqrt><mrow><mn>2</mn></mrow></msqrt></mrow></msqrt><mo>)</mo></mrow></math>. For a prime <math><mi>p</mi><mo>∤</mo><mn>2</mn><mi>a</mi></math> we obtain types of reduction (supersingular, superspecial, ordinary) of <math><msub><mrow><mi>J</mi></mrow><mrow><mi>a</mi></mrow></msub></math> at p in terms of congruences modulo 16 and the exact formulas for the zeta function of <math><msub><mrow><mi>C</mi></mrow><mrow><mi>a</mi></mrow></msub></math> over <math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math> for <math><mi>p</mi><mo>≢</mo><mn>1</mn><mo>,</mo><mn>7</mn><mrow><mo>(</mo><mi>mod</mi><mspace></mspace><mn>16</mn><mo>)</mo></mrow></math>. We deduce as conclusions the complete characterization of torsion subgroups of <math><msub><mrow><mi>J</mi></mrow><mrow><mi>a</mi></mrow></msub><mrow><mo>(</mo><mi>Q</mi><mo>)</mo></mrow></math>, namely <math><msub><mrow><mi>J</mi></mrow><mrow><mi>a</mi></mrow></msub><msub><mrow><mo>(</mo><mi>Q</mi><mo>)</mo></mrow><mrow><mi>tors</mi></mrow></msub><mo>=</mo><msub><mrow><mi>J</mi></mrow><mrow><mi>a</mi></mrow></msub><mrow><mo>(</mo><mi>Q</mi><mo>)</mo></mrow><mrow><mo>[</mo><mn>2</mn><mo>]</mo></mrow><mo>≃</mo><mi>Z</mi><mo>/</mo><mn>2</mn><mi>Z</mi></math>, and some information about <math><mi>rank</mi><mspace></mspace><msub><mrow><mi>J</mi></mrow><mrow><mi>a</mi></mrow></msub><mrow><mo>(</mo><mi>Q</mi><m

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。

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