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Author: William A. Stein
SAGE: Summer of Arithmetic Geometry Experience

# 2004 SAGE: Projects

• Verifying the Full Birch and Swinnerton-Dyer Conjecture for Specific Elliptic Curves of Analytic Rank 0 or 1
This project is at once very difficult and extremely accessible. There is a famous conjecture at the heart of the theory of elliptic curves due to the British mathematicians Bryan Birch and Sir Peter Swinnerton-Dyer. Given a specific elliptic curve over the rational numbers with "analytic rank" 0 or 1, it is in principal now possible to use the highly nontrivial theorems of Gross, Zagier, Kolyvagin, Kato, and others to verify the full conjecture for the given curve. To the best of my knowledge no systematic project has been carried out to do this for all curves in a nontrivial range. Nearly two years ago, a German computational number thoerist, Michael Stoll, approached me about carrying out such a project, and we've kicked around various ideas related to it for a while. However the ball has not started rolling. Unexpected obstructions may appear to the computations, the circumvention of which is likely to lead to new theoretical insight.

• Deciding Whether two Simple Modular Abelian Varieties are Isomorphic
My main computational research is on algorithms for computing with modular abelian varieties. A basic question when creating a computational theory for working with a class of objects (a category) is deciding whether two members of the class are isomorphic. Given suitably precise definitions, this problem is still open for modular abelian varieties, which has been a source of frustration for me in my research. There is an important special case in which I found a way to decide isomorphism, which is when the abelian varieties in question are "simple". I have not written this result up yet, except that I've partly implemented the algorithm. I'm sure the method can be made more general, but haven't had a chance to do so in any interesting cases. Tseno will hopefully learn what I've already done, help me to write it up with illustrative examples, and try to find more general methods for isomorphism testing along similar lines. He may also investigate other methods for isomorphism testing, e.g., those that are used in representation theory of finite groups. He is likely do computations with the algorithm, which will be of independent interest.

• A Vast Database of Elliptic Curves
This project is to (1) make easily available a massive collection of data about elliptic curves (much bigger than, e.g, the human genome), and (2) extract statistics from this data. This is something that many of the hundreds of people interested in elliptic curves would greatly appreciate, and simply needs to be done. Nobody has done it, and probably nobody would for a long time unless Baur does it this summer.

• Points on Elliptic Curves of Rank Bigger than One
This project is to investigate various unexplored ideas of mine and others in hopes of finding an interpretation of points on elliptic curves of rank bigger than one. Understanding points on curves of large rank is considered by some to be one of the holy grails of arithmetic geometry, and perhaps Andrei's fresh viewpoint might provide new insight. I have one challenging (but not impossible) computation that I think nobody has done, but which may shed new insight on this problem.

• Torsion Points on Elliptic Curves
About thirty years ago, Barry Mazur determined all torsion subgroups of elliptic curves over the rational numbers. It turns out that there are exactly 15 possibilities. Jennifer's project is about the same question, but over certain number fields, where the situation isn't nearly so clear. Extensive theoretical work has been done on the problem by Loic Merel and others, but it has only been answered in a few other cases. It would be far too ambitous for Jennifer to try to prove analogues of Mazur's theorem over number fields. Instead, she would likely search the literature in order to get a sense of what is, and is not, known. She might also design and coordinate computations on MECCAH that would be of interest to anyone wishing to formulate a conjecture over a given number field that generalizes Mazur's result.