CAREER: Cohomological Methods in Algebraic Geometry and Number Theory

研究者研究了在算术几何的几个方面使用p进解析技术。一个重点是在有限域上代数变量的p进上同调，包括理论问题，如系数对象在上同调操作下的稳定性，以及计算问题，如确定特定曲线和曲面的zeta函数。另一个焦点是局部域上伽罗瓦表示的分类（p-adic Hodge理论）；其目标包括深入了解伽罗瓦表示和p进李群表示之间的朗兰兹式对应关系，以及统一Hodge理论在复杂和p进环境中的处理。这些焦点共享p进微分方程理论的共同（和最近发展的）技术元素，p进微分方程也有强大的复解析类似物。在等差几何中使用p进解析方法，如在研究者的工作中，在数学中具有重要意义；例如，它目前正被用于对费马大定理证明的基础技术进行推广。但是，由于在组合学和计算机科学中有限域上多项式方程系统（如整数模质数）的出现，它也具有令人惊讶的实际意义。例如，基于椭圆曲线的加密技术已被NIST采用作为安全通信的标准，因为它在效率与安全性之间取得了平衡；p进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；可以使用p进方法来搜索能够纠正传输错误的码，而不会带来太多的开销。到目前为止，这些应用依赖于p进上同调理论中已证明的陈述，但该理论有些未完成，似乎未来的应用将依赖于尚未证明的陈述的可证明正确性。因此，理论与实践并重是十分重要的。

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.