p-adic Cohomology and Applications

Kiran Kedlaya DMS-0400272p-adic 分析奖摘要由 Hensel 在上世纪之交发起，旨在将微积分的“连续”技术应用于数论中的“离散”问题。我们研究p-adicanalysis在算术代数几何中的应用；一个典型的问题是计算多项式方程组的解。我们一方面致力于提高对这个问题的理论理解，另一方面致力于开发处理重要特殊情况的实用算法。其中一些情况发生在密码学、纠错码和计算机科学其他领域的应用中。更具体地说，我们正在与旧的 etale 上同调理论并行开发 Berthelot 的刚性上同调，后者在理论上更好理解，但不适合显式计算。一个长期目标是将 Lafforgue 在函数场 Langlands 对应方面的工作扩展到 p 进滑轮（晶体）。另一方面，我们正在研究 p 进上同调的计算算法，除了上述实际应用之外，它可能还有数学应用。

Abstract for the award of Kiran Kedlaya DMS-0400272p-adic analysis, initiated by Hensel at the turn of the last century,seeks to bring the "continuous" techniques of calculus to bear on"discrete" problems in number theory. We study applications of p-adicanalysis in arithmetic algebraic geometry; a typical problem is to countsolutions of systems of polynomial equations. We work on one hand onimproving our theoretical understanding of this problem, and on the otherhand on developing practical algorithms for treating important specialcases. Some of these cases occur in applications to cryptography, errorcorrecting codes, and other areas of computer science.More specifically, we are developing Berthelot's rigid cohomology in parallel with the older theory of etale cohomology, which is better understood theoretically but ill-equipped for explicit computations.One long-term goal is to extend Lafforgue's work on the function field Langlands correspondence to p-adic sheaves (crystals). In another direction, we are investigating algorithms for computing in p-adic cohomology, which may have mathematical applications on top of the practical ones mentioned above.