Odd moments for the trace of Frobenius and the Sato–Tate conjecture in arithmetic progressions

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications Pub Date : 2024-07-08 DOI:10.1016/j.ffa.2024.102465

Kathrin Bringmann , Ben Kane , Sudhir Pujahari

引用次数: 0

Abstract

In this paper, we consider the moments of the trace of Frobenius of elliptic curves if the trace is restricted to a fixed arithmetic progression. We determine the asymptotic behavior for the ratio of the $(2 k + 1)$ -th moment to the zeroeth moment as the size of the finite field $F_{p^{r}}$ goes to infinity. These results follow from similar asymptotic formulas relating sums and moments of Hurwitz class numbers where the sums are restricted to certain arithmetic progressions. As an application, we prove that the distribution of the trace of Frobenius in arithmetic progressions is equidistributed with respect to the Sato–Tate measure.

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算术级数中弗罗贝纽斯迹的奇矩和佐藤塔特猜想

在本文中，我们考虑了椭圆曲线 Frobenius 的迹的矩，如果迹被限制在一个固定的算术级数上。我们确定了当有限域 Fpr 的大小达到无穷大时，第 (2k+1)-th 矩与第零矩之比的渐近行为。这些结果源于赫维兹类数的和与矩的类似渐近公式，其中和被限制为某些算术级数。作为应用，我们证明了算术级数中弗罗贝尼斯迹的分布与萨托-塔特度量是等分布的。

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.