Number Theory and Related Fields

Mazur建议研究数论的统计和算法方面——特别是与椭圆曲线、阿贝尔变异、模曲线和自同构形式相关的方面。例如，他正在与卡尔·鲁宾一起研究椭圆曲线二次扭转族塞尔默群的马尔可夫统计。Selmer群为椭圆曲线和阿贝尔变的算法提供了一种精细的、可计算的规范。对塞尔默群的深入理解可以应用于广泛的问题，如可解性和不可解性问题(希尔伯特？或者精炼的解析公式（鲁宾和PI已经解决了Darmon猜想的很大一部分）。Mazur还建议继续研究模特征型及其l函数的p进插值。从广义上讲，Mazur想要理解多项式方程和可能引导我们求解这些方程的算法。研究这些具体的和统计上的解决方案，将相关的分析和算术工具用于解决问题，是非常有指导意义的。在过去，这样的研究有着令人惊讶的广泛应用——无论是在实践上还是理论上——正如PI希望的那样。

Mazur proposes to work on the statistical and algorithmic aspects of number theory---specifically as related to elliptic curves, abelian varieties, modular curves, and automorphic forms. For example, he is engaged in a research program with Karl Rubin to investigate the Markovian statistics of Selmer groups of quadratic twist families of elliptic curves. Selmer groups provide a delicate---and computable---gauge for the arithmetic of elliptic curves and abelian varieties. A close understanding of Selmer groups can be applied to questions as far-ranging as issues of solvability and unsolvability (Hilbert?s Tenth Problem) or refined analytic formulas (where Rubin and the PI have solved a good part of a conjecture of Darmon). Mazur also proposes to continue the study of p-adic interpolation of modular eigenforms and of their L-functions.In broad terms, Mazur wants to understand polynomial equations and the algorithms that might lead us to the solutions of such equations. It is vastly instructive to study such solutions, both specifically and statistically, bringing the relevant analytic and arithmetic tools to bear on the problem. Such investigations have had, in the past, surprisingly broader application---both practical and theoretical---as the PI hopes will be the case in this instance.