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

找到解多项式方程的有理数比找到实数的解更难。这是因为实数形成了一个被称为有理数的完备化的数系，而在有理数的完备化中的解通常更容易找到。人们可能希望通过仔细研究住在完备体中的更容易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。在数论中，被设计来测量这种差异的一个中心对象是Tate-Shafarevich群，而这个群只在一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。这个项目通过一种叫做岩泽理论的理论概述了这些猜想的一些进展。另一个目的是从一种现代分析形式的角度来理解特定值之间的相互作用，并利用这种相互作用来简单地解释某些晶体表示的复杂现象。在Frobenius迹消失的情况下，Wan证明了超奇异素数处椭圆曲线的岩泽主要猜想，在一般的超奇异情形下证明了PI。我们的计划是将这项工作扩展到更一般的模块化形式。在超奇异情形下的一个中心思想是构造两个适当的p次幂函数级数，当Frobenius迹为零时，这一点已被Pollack的工作明确地知道了。PI将与Otsuki合作，通过进一步发展超奇异岩泽理论的分析方面，在更一般的情况下显式地构造适当的p元幂级数对。这种显式构造的一个目标是确定晶体Galois表示的约化，揭示了Breuil的一些猜想，这些猜想纯粹根据Frobenius迹和它的Hodge-Tate重量来描述这些约化。另一个目标是建立由Bloch和Kato定义的Tate-Shafarevich群的p-初等分量大小的渐近公式，推广了雷、Loeffler和泽布斯的工作。这一奖项反映了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.