p-Adic Computation of L-Functions at Scale

最近数论的许多进展都是由于我们对固定多项式方程对变化素数模的解的个数的理解的改进而得到的。关于这一主题的猜测既可以由理论考虑驱动，也可以越来越频繁地由经验证据驱动。例如，20世纪90年代费马大定理的解决依赖于用模形式解释椭圆曲线的点计数，模形式已经以各种形式（由谷山、志村和韦尔）进行了推测，并在许多情况下得到了数值证实（由克雷莫纳）。最近，l函数和模形式数据库已经开发出来，提供了一个“数学星图”，既可以说明当前的知识，也可以作为未来猜想（然后是定理）的基础。获得该奖项支持的研究生将接受培训，为这些项目做出贡献。二十多年来，PI一直在研究技术，使机器计算算术l函数在越来越多的情况下变得实用，利用了以前认为没有算法价值的p进分析的思想。本项目的目标是将这些知识综合到一个大规模的努力中，以制表关于算术l函数的数据，用于数论、代数几何、数学物理和其他领域。我们感兴趣的一个关键案例是超几何微分方程参数化的l函数（或更一般的A-超几何系统）；除了提供各种各样的广泛兴趣的数值例子外，这些方程可以使用p进霍奇理论的最新工具（特别是移动上同调）进行研究，以期在数域设置中引入算术l函数的上同调解释。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Many recent advances in number theory have been powered by improvements in our understanding of the number of solutions of a fixed polynomial equation modulo a varying prime number. Conjectures on this topic can be driven both by theoretical considerations and, increasingly frequently, by empirical evidence. For instance, the resolution of Fermat's Last Theorem in the 1990s depended on an interpretation of point counts of elliptic curves in terms of modular forms, which had been both been conjectured in various forms (by Taniyama, Shimura, and Weil) and confirmed numerically in many instances (by Cremona). Recently, the L-Functions and Modular Forms Database has been developed to provide a "mathematical star chart" both to illustrate current knowledge and to serve as the basis for future conjectures (and then theorems). Graduate students supported by the award will receive training to contribute towards these projects.For over two decades, the PI has worked on techniques to make the machine computation of arithmetic L-functions practical in more and more cases, drawing on ideas from p-adic analysis that were not previously thought to have algorithmic value. The goal of the present project is to synthesize this knowledge into a large-scale effort to tabulate data about arithmetic L-functions for use in number theory, algebraic geometry, mathematical physics, and other areas. A key case of interest is L-functions parametrized by hypergeometric differential equations (or more generally A-hypergeometric systems); besides providing a diverse array of numerical examples of wide interest, these equations can be studied using the newest tools of p-adic Hodge theory (particularly prismatic cohomology) with an eye towards introducing cohomological interpretations of arithmetic L-functions in the number field setting.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.