Explicit Approaches to Elliptic Curves, Modular Forms and Modular Abelian Varieties

智力优势：在这个建议中的项目将推广Cremona的highlyinfluential表到下一个（按判别式排序）全实数域和高维模阿贝尔簇。这将改进椭圆曲线和阿贝尔簇的计算算法，并为数论研究者提供有用的数据和工具。本文的研究也为椭圆曲线上点的构造、椭圆曲线上的同伦类的构造以及数域上椭圆曲线的运算的理解提供了技术上的帮助。这个项目将作为一个具体的交付新的公开可用的表格和软件，将使用许多数论。更广泛的影响：PI是共同创作一个流行的书籍与BarryMazur的黎曼假设，共同创作一个研究生水平的书与Ken-neth Ribet的模形式和Hecke算子，并打算发布一个新版本的他的模形式的书。PI有可在线免费获得的数据表，其创建得到了NSF FRG资助DMS-0757627的支持，拟议的研究将进一步扩展这些表。他还将继续组织他发起的免费开源NSF资助的Sage数学软件项目的开发。PI组织了数十个“圣人日”研讨会，涉及许多本科生和研究生，涉及数论，代数拓扑，组合学，特殊函数，数值计算和其他领域。PI也是UTMOSTNSF赠款（DUE-1020378）的共同PI，其目标是使高中和大学教师和学生更容易接触Sage。

Intellectual Merit: The projects in this proposal would generalize the highlyinfluential tables of Cremona to the next (ordered by discriminant) totally realnumber field and to higher dimensional modular abelian varieties. This wouldimprove on algorithms available for computing with elliptic curves and abelianvarieties, and provide useful data and tools for number theory researchers. Theproposed research would also advance techniques for constructing points and coho-mology classes on elliptic curves, and for understanding the arithmetic of ellipticcurves over number fields. This project would have as a concrete deliverable newpublicly available tables and software that will be of use to many number theorists.Broader Impact: The PI is co-authoring a popular expository book with BarryMazur on the Riemann Hypothesis, co-authoring a graduate level book with Ken-neth Ribet on modular forms and Hecke operators, and intends to release a newedition of his modular forms book. The PI has tables of data that are freely avail-able online, and whose creation has been supported by NSF FRG grant DMS-0757627, and the proposed research would expand these tables further. He willalso continue to organize the development of the free open source NSF-fundedSage mathematical software project that he started. The PI organizes dozens of"Sage Days" workshops that involve many undergraduate and graduate students,and touch on number theory, algebraic topology, combinatorics, special functions,numerical computation, and other areas. The PI is also a co-PI on the UTMOSTNSF grant (DUE-1020378), whose goal is to make Sage more accessible to highschool and college teachers and students.