Rational Formal Groups

The paper (with Robert Coleman, nach maireann, see Robert F. Coleman 1954-2014) Pac. J Math. 47 (1991), 25--27 "Rational Formal Group Laws" gives a characterization (in characteristic zero) of 1 dimensional formal group laws that are rational functions. Local PDF copy, journal reference: Rational Formal Group Laws, Rational Formal Group Laws (PDF).

See this 2017 blog from Alexander Walker (previously in the 2013 blog entry from 'Integer Miscellany" now defunct), Formal Groups and Where to Find Them for an introduction to Formal Group Laws, with an expanded exposition of our paper, after giving a characterization of polynomial formal group laws. The author points out the results was first proved in 1976 by Robert Bismuth in an unpublished Masters Thesis at McMaster University (wrongly described in the blog post as a Doctoral thesis). Here is a direct reference: Rational Formal Group Laws (Bismuth) to the archive, and the scanned copy in PDF form is: Rational Formal Group Laws (Bismuth), (PDF). The methods used were very different, and this work was not known to me when I wrote to Coleman, suggesting his techniques could be used to solve the problem. Walker points out the method used by Bismuth works in other characteristics also.

I cannot resist pointing out --for its title as well as content-- another short earlier paper (Pac. J. Math. 122 (1986) 35--41) of Coleman: One-dimensional Algebraic Formal Groups proving "Every algebraic formal group is algebraically isomorphic to a formal algebraic group." Nowadays (2023) this provides the obvious challenge for implementation in Lean; thus providing a 'formalized' proof of these results about formal groups... To be continued!

Elliptic Curve Computations

Here is a copy of the elliptic curves site (containing tables of the curves studied) "310716 Elliptic Curves of Prime Conductor" for the Brumer-McGuinness paper from Bull. Amer. Math. Soc. (23) 1990, 375--382: The behavior of the Mordell-Weil group of elliptic curves, (PDF).

See also a copy of the site at 310716 Elliptic Curves of Prime Conductor on William Stein's web site.

Note the paper in Bull. Amer. Math. Soc. 44 (2007), 233-254, by Baur Bektemirov, Barry Mazur, William Stein and Mark Watkins, describing some extensions of these calculations, comparing our results with their more extensive calculations, together with much interesting discussion on ranks (including references to the then emerging new spectacular results of Bhargava): Average ranks of elliptic curves: Tension between data and conjecture (PDF).

In particular, see footnote 11 on page 245 for a note on some errors in our tables, details given to us by Mark Watkins, but not applied to the site.

Manjul Bhargava's Fields Medal address "Rational Points on Elliptic and HyperElliptic Curves" at the Seoul ICM in 2014 contains references to both of these papers, and uses a graph from the latter paper. See pages 657--684 of Volume I of the proceedings, downloadable here: ICM 2014 Proceedings Volume I, and watch the address on YouTube at: Rational Points on Elliptic and HyperElliptic Curves (this is preceded by the 'Laudatio' given by B.H. Gross).

Rank Bounds

TODO: Should discuss the paper of Brumer on Average Ranks of Elliptic curves here, and reference a fuller version of the Appendix by me.