Number Theory and Related Fields

马祖尔建议从事数论的统计和算法方面的工作-特别是与椭圆曲线、阿贝尔变种、模曲线和自守形式有关的工作。例如，他正在从事一项研究计划与卡尔鲁宾调查马尔可夫统计的塞尔默集团的二次扭曲家庭的椭圆曲线。塞尔默群为椭圆曲线和阿贝尔簇的算术提供了一个精细的、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（希尔伯特？的第十个问题）或精炼的解析公式（其中鲁宾和PI已经解决了很大一部分的猜想达蒙）。马祖尔还建议继续研究的p-进插值的模块特征形和他们的L-功能。在广泛的条款，马祖尔希望了解多项式方程和算法，可能会导致我们的解决方案，这样的方程。研究这些解决方案，无论是具体的还是统计的，都是非常有益的，可以把相关的分析和算术工具带到这个问题上。在过去，这样的调查有着令人惊讶的广泛应用--

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.