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

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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.