Kronecker-Weber by deformation, or: another bad reference

I have already mentioned two instances of pretty bad references in which I am involved (here and there). Here’s a third one: in Remark I.5.4 in my introductory notes on automorphic forms, L-functions and number theory (published in the proceedings of a school held at the Hebrew University in Jerusalem in March 2001), I state

Remark 1.5.4. The Kronecker-Weber Theorem, as stated here, bears a striking resemblance
to the L-function form of the modularity conjecture for elliptic curves (explained
in de Shalit’s lectures). One can prove Theorem 1.5.2 by following the general principles of
Wiles’s argument [Tu] (deformation of Galois representations, and computation of numerical
invariants in a commutative algebra criterion for isomorphism between two rings).

where the helpful-looking [Tu] leads rather disappointingly to:

[Tu] Tunnell, J.: Rutgers University graduate course (1995–96).

About a year and a half ago, R. Rhoades asked me if there was any more information available about this. The answer was that I had my own handwritten lecture notes of the original course taught by J. Tunnell at Rutgers (of course, maybe other people who had participated had their own). I said that I’d try to get those notes scanned, but it’s only in the last two days that I’ve finally started doing so — thanks to the recent installation in the ETH Library of a pretty fancy scanning machine, which makes the process essentially painless.

I’ve only scanned the first notebook and part of the second for the moment (enough to contain what Tunnell did about the GL(1) analogue of the modularity theorem of Wiles and its application to the Kronecker-Weber theorem):

Unfortunately, it’s not clear how useful these will be to anyone, except future historians of the teaching of the proof of Fermat’s Great Theorem. The lectures are not entirely linear (there are notes about a parallel seminar on Serre’s Conjecture and of a few other lectures in the middle), they are in French, and the quality of the scan is not perfect (the second notebook was particularly cheap, and the ink on one side of a page is partly visible on the other side).

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Kowalski

I am a professor of mathematics at ETH Zürich since 2008.

3 thoughts on “Kronecker-Weber by deformation, or: another bad reference”

  1. Apologies for a comment under a pseudonym.

    I was googling to try to find the original proof of Kronecker-Weber, in the form Kronecker and Weber did it before classfield theory came into picture.

    If you have a reference I will be most grateful. I didn’t have much success googling and this was one of the pages that turned up.

  2. Sorry, I don’t have a precise reference for this, but I would suggest looking at Lemmermeyer’s historical book on reciprocity laws — maybe you will find a reference there.

  3. “E. E.” may be interested in

    Neumann, Olaf
    Two proofs of the Kronecker-Weber theorem “according to Kronecker, and Weber”.
    J. Reine Angew. Math. 323 (1981), 105–126.

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