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.

研究者研究了p-Adic分析技术在算术几何形状的几个方面的使用。一个重点是代数品种在有限领域的P-ADIC共同体，包括理论问题，例如同胞操作下系数对象的稳定性以及计算问题，例如确定特定曲线和表面的Zeta功能。另一个重点是对地方领域的Galois表示（P-Adic Hodge理论）的分类；这项目标包括洞悉加洛伊斯表示与P-Adic Lie offers的langlands风格的对应关系，以及在复杂和P-ADIC环境中统一对Hodge理论的处理。这些焦点从P-ADIC微分方程理论中共有共同的（并且最近开发的）技术要素，这些方程式又具有强大的复杂分析类似物。在研究人员的工作中，使用P-ADIC分析方法在算术几何形状中具有很大的意义。例如，目前正在使用它来追求Fermat最后一个定理证明基础的技术的概括。但是，由于组合和计算机科学中多项式方程的系统（例如，整数模拟素数），这也具有令人惊讶的实际意义。例如，由于其效率与安全性的平衡，NIST采用了基于椭圆曲线的密码学作为安全通信的标准。 P-ADIC方法可用于选择适合这种结构的曲线。有限磁场上的几何形状也出现在许多错误校正代码的构建中。可以部署P-ADIC方法来搜索校正传输错误而不会携带太多开销的代码。到目前为止，这些应用依赖于P-ADIC共同体理论中的经过验证的陈述，但是该理论尚未完成，并且似乎未来的应用可能会依赖于尚未证明的陈述的可证明的正确性。因此，并行追求理论和实践方面很重要。