Relating Special Values of L-Functions with Orders of Tate-Shafarevich Groups

找到求解多项式方程的有理数比找到实数解更困难。这是因为实数形成了一个称为有理数完备化的数系，并且通常可以更容易地找到完备化中的解。人们可能希望通过仔细检查完成中的较简单的解决方案来找到硬有理数解决方案，但这两种类型的解决方案之间可能存在差异。数论中旨在衡量这种差异的一个中心对象是 Tate-Shafarevich 群，而该群只有在某些特殊情况下才能被理解。两个中心猜想（Birch 和 Swinnerton-Dyer 猜想以及 Bloch-Kato 猜想）将 Tate-Shafarevich 群的大小与适当函数的特殊值联系起来。该项目通过称为岩泽理论的理论概述了这些猜想的一些进展。另一个目标是从现代分析形式（称为 p-adic 分析）的角度理解特殊值之间的相互作用，并利用这种相互作用对某些晶体表示的复杂现象给出简单的解释。岩泽关于超奇素数椭圆曲线的主要猜想由 Wan 在 Frobenius 迹消失的情况下证明，在一般超奇异情况下 PI 也得到了证明。该计划是将这项工作扩展到更通用的模块化形式。超奇异情况的一个中心思想是构造两个适当的 p 进幂级数，当 Frobenius 迹为零时，波拉克的工作明确地知道了这一点。 PI 将与 Otsuki 合作，通过进一步发展超奇异 Iwasawa 理论的分析方面，在更一般的情况下明确构造适当的 p-adic 幂级数对。这种显式构造的目标之一是确定结晶伽罗瓦表示的约简，揭示布勒伊的一些猜想，这些猜想纯粹用弗罗贝尼乌斯迹及其霍奇-泰特权重来描述这些约简。另一个目标是建立由 Bloch 和 Kato 定义的 Tate-Shafarevich 群的 p 初级分量大小的渐近公式，概括 Lei、Loeffler 和 Zerbes 的工作。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

Finding rational numbers that solve a polynomial equation is harder than finding solutions that are real numbers. This is because the real numbers form a number system called a completion of the rational numbers, and the solutions in a completion can usually be found more easily. One may hope to find the hard rational number solutions by scrutinizing the easier solutions living in the completions, but there may be a discrepancy between these two types of solutions. One central object in number theory designed to measure such a discrepancy is the Tate-Shafarevich group, and this group is only understood in some special cases. Two central conjectures (the Birch and Swinnerton-Dyer conjecture and the Bloch-Kato conjecture) relate sizes of Tate-Shafarevich groups to special values of an appropriate function. This project outlines some progress on these conjectures via a theory called Iwasawa theory. Another goal is to understand the interplay between the special values from the perspective of a modern form of analysis, called p-adic analysis, and use this interplay to give an easy explanation of complicated phenomena of certain crystalline representations.The Iwasawa main conjecture for elliptic curves at supersingular primes was proved by Wan in the case in which the trace of Frobenius vanishes, and the PI in the general supersingular case. The plan is to extend this work to more general modular forms. One central idea in the supersingular case is the construction of two appropriate p-adic power series, which is known explicitly by work of Pollack when the Frobenius trace is zero. The PI will work with Otsuki to explicitly construct the appropriate pair of p-adic power series in more general cases by developing the analytic aspect of supersingular Iwasawa theory further. One goal of such an explicit construction is determining reductions of crystalline Galois representations, shedding light on some conjectures of Breuil which describe these reductions purely in terms of the Frobenius trace and its Hodge-Tate weight. Another goal is to establish asymptotic formulas for the size of the p-primary components of Tate-Shafarevich groups as defined by Bloch and Kato, generalizing work of Lei, Loeffler, and Zerbes.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.