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分类一般如下：（a）算术和几何的阿贝尔品种：在早期的工作中，我们发起了一个新的局部全局问题，涉及简单的阿贝尔品种超过一个数域的因素时，减少模一个素数。这个问题的动机是密码学，但即使是纯粹的算术代数几何问题也很有趣。它的推广可以用Tate圈和“几乎代数”圈来表示。我们将使用Tannakian形式主义来研究这些。(b)模形式和L-函数：我们将研究佐藤-泰特猜想中的误差项。我们还将研究欧拉-克罗内克常数相关联的（由Ihara），以一个数字领域的增长性质（解决一些问题提出的Ihara），以及其算术性质。(c)信息技术：在早期的工作中，我们考虑了Lehmer猜想的一个变体，关于Ramanujan tau函数的消失。我们的变体涉及正整数n和tau函数在n处的值之间的公因子。在这项工作的基础上，以及后来的工作与S。枪和N。Laptyeva和A.周，我们将继续我们的调查，这些结果，也发展的作用，模形式可以发挥因式分解算法。我们还将继续研究基于流的数据完整性算法。