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

找到解决多项式方程的有理数比找到真实的数的解更难。这是因为真实的数形成了一个称为有理数完备化的数字系统，而完备化中的解通常更容易找到。人们可能希望通过仔细检查存在于完备中的较容易的解来找到硬有理数解，但是这两种类型的解之间可能存在差异。数论中用来度量这种差异的一个中心对象是泰特-沙法列维奇群，这个群只在某些特殊情况下才被理解。两个中心猜想（Birch和Swinnerton-Dyer猜想和Bloch-Kato猜想）将Tate-Shafarevich群的大小与适当函数的特殊值联系起来。该项目通过一种称为岩泽理论的理论概述了这些理论的一些进展。另一个目标是从现代分析形式的角度理解特殊值之间的相互作用，称为p-adic分析，并使用这种相互作用来简单地解释某些晶体表示的复杂现象。岩泽主要猜想椭圆曲线在超奇异素数的情况下被Wan证明，其中Frobenius的踪迹消失，而PI在一般的超奇异情况下。计划将这项工作扩展到更一般的模块化形式。超奇异情形的一个中心思想是构造两个适当的p-adic幂级数，当Frobenius迹为零时，Pollack的工作明确地知道这一点。PI将与大月合作，通过进一步发展超奇异岩泽理论的分析方面，在更一般的情况下明确地构造适当的p进幂级数对。这样一个明确的建设的一个目标是确定减少结晶伽罗瓦表示，揭示了一些abstractures的Breuil描述这些减少纯粹的弗罗贝纽斯跟踪和其霍奇-泰特重量。另一个目标是建立由Bloch和Kato定义的Tate-Shafarevich群的p-主分量的大小的渐近公式，推广Lei，Loeffler和Zerbes的工作。该奖项反映了NSF的法定使命，并被认为值得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来支持。

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.