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At the DeKalb conference on Hilbert's problems, John Tate gave a masterly survey of Problem 9, the General Reciprocity Law. He ends with a discussion of the Langlands Programme, especially the case of odd Artin representations $R$ of degree $2$. Let me quote from the final paragraph of the written version of his talk :

Another reason for the relationship's eluding Artin and Hecke may be the fact that explicit non-dihedral numerical examples are hard to find. Indeed at the time of the DeKalb conference, none was known ! I concluded the oral presentation of the paper there by explaining that, in the hope of finding one, I had looked for non-dihedral $R$'s of low conductor, $N$, and had found an $R$ with $N=133=7\cdot19$ which I hoped might be amenable to computation. After the talk, Atkin suggested that the labor involved might be considerably reduced by systematic use of $w_7$ and $w_{19}$. Armed with his theory of the $w$'s, four Harvard students, D. Flath, R. Kottwitz, J. Tunnell, J. Weisinger and I succeeded in the next month in proving (by relatively easy hand computation) the existence of the corresponding new form $f_R$ of weight $1$ and level $133$ predicted by Langlands.

My first question is : what was $R$ ? The second question is : how would you verify today (on a computer) the existence of $f_R$, without invoking any of the theorems which have been proved in the meantime ?

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1 Answer 1

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$R$ is a 2-dimensional conductor 133 representation of the absolute Galois group of the rationals into $GL(2,\mathbf{C})$, whose associated representation to $PGL(2,\mathbf{C})$ cuts out the $A_4$ extension of the rationals which is the splitting field of $x^4 + 3x^2 - 7x + 4$. To verify the existence of the weight 1 form of level 133 you can just fire up a magma session (magma is a computer algebra package) and ask it to compute the weight 1 level 133 forms, and the dihedral weight 1 level 133 forms, and then note that there are more weight 1 level 133 forms than dihedral ones. So it's pretty easy now. Or you can write such programs yourself! [which is what I did and which is why magma can do it ;-) ]. More shameless self-promotion available (including the algorithm) at

http://www2.imperial.ac.uk/~buzzard/maths/research/papers/wt1.pdf

(where I e.g. explain that there's even a level 124 non-dihedral form, and give a description of the $S_4$ form of smallest conductor). However the ideas all first appeared in print in Joe Buhler's thesis donkey's years ago, where he finds an $A_5$ form: Springer LNM654.

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Great ! It's good that I asked the question here instead of just looking up Buhler's LNM volume: the paper you link to is only two weeks old. –  Chandan Singh Dalawat May 22 '12 at 7:17
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Chandan -- I knew that Tate had discovered an $A_4$ example years ago -- this fact is mentioned in Buhler's thesis if I remember correctly -- and I did chase up Tate's paper when I was a grad student. But it was foolish of me to not chase it up again. I've added some historical remarks plus the reference to Tate's paper, to my paper. Thanks. Amazing example of MO working to make life better. –  Kevin Buzzard May 22 '12 at 10:38
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It took me a while to figure out what a 'thesis donkey' was. –  Stopple May 22 '12 at 15:44
    
@Stopple: I'm sorry I don't get it; what is a "thesis donkey"? –  Giuseppe May 23 '12 at 17:32
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Oh dear. Is this a Britishism? I was once giving a talk in Berkeley and I described something as "wonky" and someone in the audience asked me what it meant -- I was surprised that this word could be a British English word but not a US English one. So I told them that "wonky" meant "skewiff" and they said that they'd never heard of that word either! –  Kevin Buzzard May 24 '12 at 14:47

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