p-Adic Computation of L-Functions at Scale

点击翻译按钮获取中文摘要

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.