Everyone Seminar, October 1993

The official notes on my October 22, 1993 Albany "Everyone Seminar" presentation give a beginning introduction to the final resolution of Fermat's Last Theorem through the 1993 work of Andrew Wiles on the modular curve conjecture following the work of Frey, Serre, and Ribet in the mid 1980's.

Since 1994 I have been looking for a truly reliable way to make mathematical materials available through the web without the need for editing more than one source. This led to my GELLMU Project. In February 2001 I re-edited the notes under the new system so that, as of that time, there was an HTML version for the benefit of (1) those sitting where neither DVI nor PDF was available and (2) those who could not use the typeset formats, including web crawlers. Since the fall of 2004, it has been possible to generate automatically from the same source a fully reliable web version making use of Mathematical Markup Language (MathML).

The new edition is available in the following formats:

1. HTML (version 5) with MathJax rendering.
2. XML form of HTML extended by MathML.
3. Classical HTML.
4. PDF for paged viewing and for printing. (A screen version, which would be formatted differently and would include anchors, is not presently available.)
5. DVI for paged viewing and printing.
6. LaTeX translation.
7. author-level XML, the base of all output formattings.
8. GELLMU source, the only source, from which everything else was automatically derived.

The original notes are available in the following formats:

The notes, which were completed in March 1994, contain as an appendix a public announcement by Andrew Wiles about an unfinished argument. Subsequent to that time in October 1994 Karl Rubin posted an announcement at Ohio State, which received wide electronic circulation, and that appeared to clear the air. The papers advertised by Rubin appeared in the Annals of Mathematics issue dated May 1995.

As of the Annals articles, the Shimura-Taniyama-Weil conjecture had been proved only for semi-stable elliptic curves. (This settled the celebrated corollary.) It left open for general elliptic curves defined over the rational field, as well as for much more general objects that are realizable as schemes of finite type over the ring of integers, the question of whether the zeta function behaves as a zeta function should and as the zeta function of the "final" scheme over the ring of integers does.

By the summer of 1999 there were widespread reports that the full Shimura-Taniyama-Weil conjecture (or the modular curve conjecture), covering all elliptic curves defined over the rational field, had been proved by Breuil, Conrad, Diamond, and Taylor.

The following is a short list of post 1993 references:

1. C. Breuil, B. Conrad, F. Diamond, & R. Taylor, On the modularity of elliptic curves over Q: wild 3-adic exercises'', J. Amer. Math. Soc., to appear. See http://www.math.harvard.edu/~rtaylor/.
2. J. Coates & S.T. Yau, Elliptic curves, modular forms, & Fermat's last theorem, 2nd edition, International Press, Cambridge, MA, 1997. Proceeding of the Conference on Elliptic Curves and Modular Forms held at the Chinese University of Hong Kong, Dec. 1993.
3. G. Cornell, J. H. Silverman, & G. Stevens, Modular Forms and Fermat's Last Theorem, Springer-Verlag, 1997. This volume is the record of an instructional conference on number theory and arithmetic geometry held August 9-18, 1995 at Boston University.
4. H. Darmon, F. Diamond, & R. Taylor, ''Fermat's Last Theorem'', Current Developments in Mathematics, 1995, International Press, Cambridge, Massachusetts, 1995.
5. R. Taylor & A. Wiles, Ring-theoretic properties of certain Hecke algebras'', Annals of Mathematics, (second series) vol. 141 (1995), pp. 553-572.
6. A. Wiles, Modular elliptic curves and Fermat's Last Theorem'', Annals of Mathematics, (second series) vol. 141 (1995), pp. 443-551.

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