Arithmetic Algebraic Geometry

我们将研究以下三个问题：(a)阿贝尔变体的算术和几何：在早期的工作中，我们提出了一个新的局部-全局问题，涉及数域上的简单阿贝尔变体，当对素数进行约模时。这个问题是由密码学引起的，但即使作为一个纯粹的算术代数几何问题，它也很有趣。它的推广可以用Tate环和“几乎代数”环来表示。我们将使用坦纳基的形式主义来研究这些问题。(b)模形式与l -函数：我们将研究Sato-Tate猜想中的误差项。我们还将研究（由Ihara提出的）与数域相关的欧拉-克罗内克常数的增长性质（解决Ihara提出的一些问题）和算术性质。(c)信息技术：在早期的工作中，我们考虑了Lehmer关于Ramanujan tau函数消失的猜想的一种变体。我们的变体涉及正整数n和n处的tau函数值之间的公因子。基于这项工作，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，我们将继续研究这些结果，并发展模形式在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。

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.