p-adic Methods in Number Theory

本奖项为参加2015年5月26-30日在加州大学伯克利分校举行的“数论中的p进方法”会议提供支持。自从库尔特·亨塞尔在1900年左右提出p进数的概念以来，p进数在数论中发挥了核心作用；例如，它们在费马大定理的证明中起到了关键作用。对数论家来说，p进数和实数一样“实数”，而且同样重要。这两种方法都是“填补”只考虑有理数所留下的空白。加藤、黑川和齐藤在他们的著作《数论I：费马的梦》中诗意地写道：“在漫长的数学历史中，数字意味着实数，直到最近我们才意识到有一个p进数的世界。这就像那些只在白天见过天空的人对夜空感到惊奇一样。[]就像我们在晚上能更清楚地看到太空物体一样，我们开始通过p进数看到深奥的数学宇宙。”这次会议将汇集p进数及其应用的许多不同方面的专家，将促进各种数论学家之间思想的交流，将使研究生和博士后接触到最先进的技术和结果，并将促进未被充分代表的少数民族和妇女参与高级数论研究。关于数论中p进方法的会议是及时和重要的，因为最近许多引人注目的数论进展都使用了深度p进方法。例如，我们提到了最近建立p进局部朗兰兹对应的特殊情况的工作；绝大多数奇次超椭圆曲线只有一个有理点的证明非阿贝尔Coleman积分和曲线上积分点的研究进展p进霍奇理论基本曲线的研究；以及最近关于完美曲面空间的研究结果。更多信息请访问https://sites.google.com/site/padicmethods2015/。

This award provides support for participation in the conference "p-adic Methods in Number Theory" held at the University of California, Berkeley on May 26-30, 2015. Since their conception by Kurt Hensel around 1900, p-adic numbers have played a central role in number theory; for example, they are used in a crucial way in the proof of Fermat's Last Theorem. To a number theorist, p-adic numbers are just as "real" -- and just as important -- as real numbers. Both are ways of "filling in the gaps" left by considering just rational numbers. In their book "Number Theory I: Fermat's Dream," Kato, Kurokawa, and Saito write poetically, "In the long history of mathematics a number meant a real number, and it is only relatively recently that we realized that there is a world of p-adic numbers. It is as if those who had seen the sky only during the day are marveling at the night sky. [ ] Just as we can see space objects better at night, we begin to see the profound mathematical universe through the p-adic numbers." This conference will bring together experts in the many different facets of p-adic numbers and their applications, will promote a cross-fertilization of ideas between number theorists of all stripes, will expose graduate students and postdocs to state-of-the-art techniques and results, and will promote participation by underrepresented minorities and women in high-level number theory research. A conference on p-adic methods in number theory is timely and important, as many spectacular recent number-theoretic advances have made use of deep p-adic methods. We mention, for example, recent work establishing special cases of the p-adic local Langlands correspondence; the proof that most hyperelliptic curves of odd degree have just one rational point; developments on non-abelian Coleman integration and integral points on curves; work on the fundamental curve of p-adic Hodge theory; and recent results on perfectoid spaces. More Information can be found at https://sites.google.com/site/padicmethods2015/.