Arithmetic Algebraic Geometry

算术代数几何

• 批准号: 44342-2013
• 负责人: Murty, Vijayakumar
• 金额: $ 2.77万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635516
• 关键词: Arithmetic; Algebraic; Geometry

We shall study three problems classified generally as follows:(a) Arithmetic and Geometry of Abelian Varieties: In earlier work, we initiated a new local-global problem involving simple Abelian varieties over a number field that factor when reduced modulo a prime. The question is motivated by cryptography, but is of interest even as a question purely in arithmetic algebraic geometry. A generalization of it can be expressed in terms of Tate cycles and 'almost algebraic' cycles. We shall investigate these using a Tannakian formalism.(b) Modular Forms and L-functions: We shall study error terms in the Sato-Tate conjecture. We shall also study the Euler-Kronecker constant associated (by Ihara) to a number field both in terms of its growth properties (addressing some questions raised by Ihara) as well as its arithmetic properties.(c) Information Technology: In earlier work, we considered a variant of Lehmer's conjecture on the vanishing of the Ramanujan tau function. Our variant concerned common factors between a positive integer n and the value of the tau function at n. Building on this work, as well as later work with S. Gun and N. Laptyeva, and the ongoing thesis work of A. Chow, we shall continue our investigation of these results, and also develop the role that modular forms can play in factorization algorithms. We shall also continue our investigations into stream-based algorithms for data integrity.

我们将研究一般分类如下的三个问题： (a) 阿贝尔簇的算术和几何：在早期的工作中，我们提出了一个新的局部-全局问题，涉及一个数域上的简单阿贝尔簇，该问题对素数进行约简分解。这个问题是由密码学引发的，但即使作为一个纯粹的算术代数几何问题也很有趣。它的概括可以用泰特循环和“几乎代数”循环来表达。我们将使用 Tannakian 形式主义来研究这些。(b) 模形式和 L 函数：我们将研究 Sato-Tate 猜想中的误差项。我们还将研究与数域相关的欧拉-克罗内克常数（由 Ihara 提出），包括其增长特性（解决 Ihara 提出的一些问题）以及算术特性。 (c) 信息技术：在早期的工作中，我们考虑了莱默关于拉马努扬 tau 函数消失的猜想的一个变体。我们的变体涉及正整数 n 和 tau 函数在 n 处的值之间的公因数。基于这项工作，以及后来与 S. Gun 和 N. Laptyeva 的合作，以及 A. Chow 正在进行的论文工作，我们将继续研究这些结果，并开发模块化形式在因式分解算法中可以发挥的作用。我们还将继续研究基于流的数据完整性算法。