Abandonware

So I am all in favor of avoiding publishing all but a select number of papers if you can help it, and blogging about math instead. So take a spoon, pass around the brandy butter and plum pudding, and, for the rest of this post, let us tuck in to something from the apocrypha.

Galois Extensions Unramified Away From One Place:

I learned about one version of this question in the tea room at Havard from Dick Gross. Namely, does there exist a non-solvable Galois extension K/Q unramified at all primes except p? Modular forms (even just restricting to the two eigenforms of level one and weights 12 and 16) provide a positive answer for p greater than 7. On the other hand, Serre’s conjecture shows that this won’t work for the last three remaining primes. Dick explained a natural approach for the remaining primes, namely to consider instead Hilbert modular forms over a totally real cyclotomic extension ramified at p (once you work out how to actually compute such beasts in practice). And indeed, this idea was successfully used to find such representations by Lassina Dembélé in this paper and also this paper (with Greenberg and Voight). But there is something a little unsatisfactory to me about this, namely, these extensions are all ramified at and What if one instead asks Gross’ question for a single place?

Minkowski showed there are no such extensions when but I don’t see any obstruction to there being a positive answer for a finite place. The first obvious remark, however, is that Galois representations coming from Hilbert modular forms are not going to be so useful in this case at least when the residual characteristic is odd, for parity reasons.

On the other hand, conjecturally, the Langlands program still has something to say about this question. One could ask, for example, for the smallest prime p for which there exists a Galois representation:

whose image is big (say not only irreducible but also not projectively exceptional) and is unramified at all places away from p including infinity. (This is related to my first ever blog post.) Here is how one might go about finding such a representation, assuming the usual suite of conjectures. First, take an imaginary quadratic field F, and then look to see if there is any extra mod-p cohomology of in some automorphic local system which is not coming from any of the “obvious” sources. If you find such a class, you could then try to do the (computationally difficult) job of computing Hecke eigenvalues, or alternatively you could do the same computation for a different such imaginary quadratic field E, and see if you find a weight for which there is an “interesting” class simultaneously for both number fields. If there are no such classes for any of the (finitely many) irreducible local systems modulo p, then there are (conjecturally) no Galois representations of the above form.

There are some heuristics (explained to me by Akshay) which predict that the number of Galois representations of the shape we are looking for (ignoring twists) is of the order of 1/p. On the other hand, no such extensions will exist for very small p by combining an argument of Tate together with the Odlyzko bounds. So the number of primes up to X for which there exist such a representation might be expected to be of the form

for some constant C to account for the lack of small primes (which won’t contribute by Tate + Odlyzko GRH discriminant bounds). This is unfortunately a function well-known to be constant, and in this case, with the irritating correction term, it looks pretty much like the zero constant. Even worse, the required computation becomes harder and harder for larger p, since one needs to compute the cohomology in the corresponding local system of weight for k up to (roughly) p. Alas, as it turns out, these things are quite slippery:

Lemma: Suppose is absolutely irreducible with Serre level 1 and Serre weight k and is even. Assume all conjectures. Then:

1. The prime is at least 79.
2. The weight is at least 33.
3. If exists with then
4. If exists with then or and is the unique representation with projective image

Of course the extension for (which is well-known) does not have big image in the sense described above.
The most annoying thing about this computation (which is described in the apocrypha) is that it can only be done once! Namely, someone who could actually program might be able to extend the computation to (say) but the number of extensions which one would expect to see is roughly which is smaller than a fifth. So maybe an extension of this kind will never be found! (Apologies for ruining it by not getting it right the first time.)