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.

我们将主要研究三个问题：(A)Abelian簇的算术和几何：在以前的工作中，我们在一个数域上提出了一个新的局部-整体问题，涉及到模为素数时的因子的简单Abel簇。这个问题的动机是密码学，但即使作为一个纯粹的算术代数几何的问题也是有意义的。它的一个推广可以用泰特圈和‘几乎代数’圈来表示。我们将使用Tannakian形式论来研究这些问题。(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(由Ihara提出的)与数域相关的Euler-Kronecker常数的增长性质(解决了Ihara提出的一些问题)以及它的算术性质。(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，即Ramanujan tau函数的消失。我们的变体涉及正整数n和n处tau函数的值之间的公因子。在这项工作以及后来与S.Gun和N.Laptyeva的工作以及A.Chow正在进行的论文工作的基础上，我们将继续研究这些结果，并开发模形式在因式分解算法中所起的作用。我们还将继续研究基于流的数据完整性算法。