Thursday, October 30, 2008

Verification of the Birch and Swinnerton-Dyer Conjecture for Specific Elliptic Curves

I spent last night and this morning making changes to my paper Verification of the Birch and Swinnerton-Dyer Conjecture for Specific Elliptic Curves. I had been sitting on the referee report for several months, since there are four co-authors, and I naively hoped one of them would fix all the problems.

This is a paper I wrote nearly four years ago really as a summer student project: SAGE -- Summer of Arithmetic Geometry Experience -- that I ran at Harvard for a group of undergrads. Five undergrads there all independently requested to work with me on research over the summer, and this paper was one of the results of that summer.

This paper was also the first application of the Sage Mathematics Software. I tried to repeat some calculations from the paper when making improvements suggested by the referee, and was scared that I would find bugs that prevent me from doing them. Indeed, within a few seconds I was disappointed to find a showstopper bug, which fortunately I was able to fix in a few minutes.

The referee remarks about the paper were almost all very useful; they definitely improve the quality of the paper a lot.

The paper itself proves most of the BSD conjectural formula for elliptic curves of conductor up to 1000. The obvious next steps with that paper done are:

  1. Mostly prove BSD for far more curves (say up to conductor 130,000). This would be subtle and interesting, since all kinds of problems that don't quite happen for conductor up to 1000 would certainly happen here.
  2. Completely prove BSD for all curves up to conductor 1,000, except the 18 curves of rank 2. This will rely on my joint paper with Wuthrich, which will make this almost completely straightforward, except perhaps in a handful of surprising cases.
  3. Prove the prediction of BSD about Sha at a bunch of primes for curves of rank 2. This can also be done using my joint paper with Wuthrich.

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