Number Theory and Related Fields

我建议研究模特征形式的p进插值，以及算术中的统计问题，例如，最近在佐藤-塔特猜想中的进展；我将继续一个长期项目，与卡尔·鲁宾合作，研究塞尔默群；最近我们一直在把我们的工作应用到各个方面。例如，为了获得数学逻辑中的一些结果（我们证明了经典shafareich - tate猜想的一个部分意味着希尔伯特的第十问题（Hilbert’s 10th Problem），该问题适用于在有理数环上生成的任意非整数的可交换环，并且有一个负的答案），并解决了关于非整数环的Darmon猜想。经典“解析公式”的内德形式狄利克雷l函数在零处的一阶导数，以及研究二阶扭转族中2-Selmer秩等于给定数的椭圆曲线的比例。我们的目标是在这些方向上取得进展。更广泛的问题是我上面所描述的建议的重点是经典问题：确定多项式方程的解的结构；描述无法通过算法解决的算术问题；而研究数据的统计特征对学习算术是很重要的。

I propose to work on p-adic interpolation of modular eigenforms, and on statistical issues in arithmetic arising from, for example, the recent advances in the conjecture of Sato-Tate; I will continue a long-range project, joint with Karl Rubin, to study Selmer groups; recently we have been applying our work in various directions. For example, to obtain some results in mathematical logic (we show that a piece of the classical Shafarevich-Tate Conjecture implies that Hilbert's Tenth Problem phrased for any commutative ring of infinite cardinality that is finitely generated over the ring of rational integers has a negative answer) and to resolve a conjecture of Darmon regarding a refined form of the classical ''analytic formula'' for the first derivative at zero of a Dirichlet L-function, and to study the proportions, in a family of quadratic twists, of the elliptic curves with 2-Selmer rank equal to a given number. Our goal is to make advances in each of these directions.The broader issues that are the focus of the aspects of my proposal described above are the classical ones: determining the structure of solutions to polynomial equations; of delineating arithmetic problems that admit of no algorithmic solution; and studying the statistical features of data important to the study of arithmetic.