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.

我们将研究三个问题，通常分类如下：（a）算术和阿贝尔品种的几何形状：在较早的工作中，我们启动了一个新的局部全球问题，其中涉及简单的Abelian品种在减少Modulo a Prime时的数字字段上。这个问题是由密码学动机的，但即使是一个纯粹是在算术代数几何形状中的问题也是一个问题。它的概括可以用泰特周期和“几乎代数”循环表示。我们将使用Tannakian形式主义进行调查。（b）模块化形式和l功能：我们将在萨托特猜想中研究误差术语。我们还将研究Euler-Kronecker常数（由Ihara）与数字领域相关联（解决Ihara提出的一些问题）及其算术特性。（c）信息技术：在较早的工作中，我们认为Lehmer对Ramanujan Tau的消失的猜测是一种变异。我们的变体涉及正整数n与n处tau函数的值之间的共同因素。在这项工作的基础上，并随后与S. Gun和N. Laptyeva一起工作，以及A. Chow的持续论文工作，我们将继续对这些结果进行调查，并发展模块化形式可以在分解算法中发挥作用的作用。我们还将继续研究基于流的数据完整性算法。