CAREER: Cohomological Methods in Algebraic Geometry and Number Theory

职业：代数几何和数论中的上同调方法

• 批准号: 0545904
• 负责人: Kiran Kedlaya
• 金额: $ 40万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-09-01 至 2012-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0545904&HistoricalAwards=false
• 关键词: CAREER; Cohomological; Methods; Algebraic; Geometry

The investigator studies the use of p-adic analytic techniques in several aspects of arithmetic geometry. One focus is on the p-adic cohomology of algebraic varieties over finite fields, including theoretical questions like the stability of coefficient objects under cohomological operations, and computational problems like the determination of zeta functions of specific curves and surfaces. Another focus is the classification of Galois representations over local fields (p-adic Hodge theory); goals of this include gaining insight into proposed Langlands-style correspondences between Galois representations and representations of p-adic Lie groups, as well as unifying the treatment of Hodge theory in the complex and p-adic settings. These foci share common (and recently developed) technical elements from the theory of p-adic differential equations, which again have strong complex-analytic analogues.The use of p-adic analytic methods in arithmetic geometry, as in the investigator's work, has great significance within mathematics; for instance, it is currently being used to pursue generalizations of the techniques underlying the proof of Fermat's Last Theorem. But it also has surprising practical significance due to the appearance of systems of polynomial equations over finite fields (such as the integers modulo a prime number) in combinatorics and computer science. For instance, elliptic curve-based cryptography has been adopted by NIST as a standard for secure communications thanks to its balance of efficiency versus security; p-adic methods can be used to select curves suitable for this construction. Geometry over finite fields also appears in the construction of many error-correcting codes; p-adic methods can be deployed to search for codes which correct transmission errors without carrying too much overhead. So far these applications rely on proven statements in the theory of p-adic cohomology, but the theory is somewhat unfinished and it seems likely that future applications will rely for their provable correctness on statements not yet proved. It is thus important to pursue theoretical and practical aspects in parallel.

调查研究使用p-adic分析技术在算术几何的几个方面。一个重点是有限域上代数簇的p进上同调，包括理论问题，如上同调运算下系数对象的稳定性，以及计算问题，如确定特定曲线和曲面的zeta函数。另一个重点是局部域上伽罗瓦表示的分类（p-adic霍奇理论）;其目标包括深入了解伽罗瓦表示和p-adic李群表示之间的朗兰兹式对应，以及统一霍奇理论在复和p-adic设置中的处理。这些焦点共享共同的（和最近开发的）技术元素从理论的p-adic微分方程，其中再次有强大的复解析analogs.The使用的p-adic分析方法在算术几何，在调查员的工作，有很大的意义在数学;例如，它目前正在被用来追求推广的技术基础的证明费马大定理。但它也有令人惊讶的实际意义，由于出现系统的多项式方程有限域（如整数模素数）在组合学和计算机科学。例如，基于椭圆曲线的密码学已经被NIST采用作为安全通信的标准，这要归功于它在效率与安全性之间的平衡; p-adic方法可以用来选择适合这种构造的曲线。有限域上的几何也出现在许多纠错码的构造中; p-adic方法可以被部署来搜索纠正传输错误而不携带太多开销的代码。到目前为止，这些应用程序依赖于证明陈述的理论p-adic上同调，但该理论是有点未完成的，它似乎可能是未来的应用程序将依赖于其可证明的正确性的陈述尚未证明。因此，必须同时进行理论和实践方面的工作。