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Fox-Neuwirth-Fuks cells, quantum shuffle algebras, and Malle’s conjecture for function fields: a new old paper

I have a new paper up on the arXiv today with TriThang Tran and Craig Westerland, “Fox-Neuwirth-Fuks cells, quantum shuffle algebras, and Malle’s conjecture for function fields.”

There’s a bit of a story behind this, but before I tell it, let me say what the paper’s about. The main result is an upper bound for the number of extensions with bounded discriminant and fixed Galois group of a rational function field F_q(t). More precisely: if G is a subgroup of S_n, and K is a global field, we can ask how many degree-n extensions of K there are whose discriminant is at most X and whose Galois closure has Galois group G. A long-standing conjecture of Malle predicts that this count is asymptotic to c X^a (log X)^b for explicitly predicted exponents a and b. This is a pretty central problem in arithmetic statistics, and in general it still seems completely out of reach; for instance, Bhargava’s work allows us to count quintic extensions of Q, and this result was extended to global fields of any characteristic other than 2 by Bhargava, Shankar, and Wang. But an asymptotic for the number of degree 6 extensions would be a massive advance.

The point of the present paper is to prove upper bounds for counting field extensions in the case of arbitrary G and rational function fields K = F_q(t) with q prime to and large enough relative to |G|; upper bounds which agree with Malle’s conjecture up to the power of log X. I’m pretty excited about this! Malle’s conjecture by now has very robust and convincing heuristic justification, but there are very few cases where we actually know anything about G-extensions for any but very special classes of finite groups G. There are even a few very special cases where the method gives both upper and lower bounds (for instance, A_4-extensions over function fields containing a cube root of 3.)

The central idea, as you might guess from the authors, is to recast this question as a problem about counting F_q-rational points on moduli spaces of G-covers, called Hurwitz spaces; by the Grothendieck-Lefschetz trace formula, we can bound these point counts if we can bound the etale Betti numbers of these spaces, and by comparison between characteristic p and characteristic 0 we can turn this into a topological problem about bounding cohomology groups of the braid group with certain coefficients.

Actually, let me say what these coefficients are. Let c be a subset of a finite group G closed under conjugacy, k a field, and V the k-vectorspace spanned by c. Then V^{\otimes n} is spanned by the set of n-tuples (g_1, … , g_n) in c^n, and this set carries a natural action of the braid group, where twining strand i past strand i+1 corresponds to the permutation

(g_1, \ldots, g_n) \rightarrow (g_1, \ldots, g_{i+1}, g_{i+1}^{-1} g_i g_{i+1}, \ldots, g_n).

So for each n we have a representation of the braid group Br_n, and it turns out that everything we desire would be downstream from good bounds on

\dim H^i(Br_n, V^{\otimes n})

So far, this is the same strategy (expressed a little differently) than was used in our earlier paper with Akshay Venkatesh to get results towards the Cohen-Lenstra conjecture over F_q(t). That paper concerned itself with the case where G was a (modestly generalized) dihedral group; there was a technical barrier that prevented us from saying anything about more general groups, and the novelty of the present paper is to find a way past that restriction. I’m not going to say very much about it here! I’ll just say it turns out that there’s a really nice way to package the cohomology groups above — indeed, even more generally, whenever V is a braided vector space, you have these braid group actions on the tensor powers, and the cohomology groups can be packaged together as the Ext groups over the quantum shuffle algebra associated to V. And it is this quantum shuffle algebra (actually, mostly its more manageable subalgebra, the Nichols algebra) that the bulk of this bulky paper studies.

But now to the story. You might notice that the arXiv stamp on this paper starts with 17! So yes — we have claimed this result before. I even blogged about it! But… that proof was not correct. The overall approach was the same as it is now, but our approach to bounding the cohomology of the Nichols algebra just wasn’t right, and we are incredibly indebted to Oscar Randall-Williams for making us aware of this.

For the last six years, we’ve been working on and off on fixing this. We kept thinking we had the decisive fix and then having it fall apart. But last spring, we had a new idea, Craig came and visited me for a very intense week, and by the end I think we were confident that we had a route — though getting to the present version of the paper occupied months after that.

A couple of thoughts about making mistakes in mathematics.

  • I don’t think we really handled this properly. Experts in the field certainly knew we weren’t standing by the original claim, and we certainly told lots of people this in talks and in conversations, and I think in general there is still an understanding that if a preprint is sitting up on the arXiv for years and hasn’t been published, maybe there’s a reason — we haven’t completely abandoned the idea that a paper becomes more “official” when it’s refereed and published. But the right thing to do in this situation is what we did with an earlier paper with an incorrect proof — replaced the paper on arXiv with a placeholder saying it was inaccurate, and issued a public announcement. So why didn’t we do that? Probably because we were constantly in a state of feeling like we had a line on fixing the paper, and we wanted to update it with a correct version. I don’t actually think that’s a great reason — but that was the reason.
  • When you break a bone it never exactly sets back the same way. And I think, having gotten this wrong before, I find it hard to be as self-assured about it as I am about most things I write. It’s long and it’s grainy and it has a lot of moving parts. But we have checked it as much as it’s possible for us to check it, over a long period of time. We understand it and we think we haven’t missed anything and so we think it’s correct now. And there’s no real alternative to putting it out into the world and saying we think it’s correct now.
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Fox-Neuwirth-Fuks cells, quantum shuffle algebras, and Malle’s conjecture for function fields

I’ve gotten behind on blogging about preprints!  Let me tell you about a new one I’m really happy with, joint with TriThang Tran and Craig Westerland, which we posted a few months ago.

Malle’s conjecture concerns the number of number fields with fixed Galois group and bounded discriminant, a question I’ve been interested in for many years now.  We recall how it goes.

Let K be a global field — that is, a number field or the function field of a curve over a finite field.  Any degree-n extension L/K (here L could be a field or just an etale algebra over K — hold that thought) gives you a homomorphism from Gal(K) to S_n, whose image we call, in a slight abuse of notation, the Galois group of L/K.

Let G be a transitive subgroup of S_n, and let N(G,K,X) be the number of degree-n extensions of K whose Galois group is G and whose discriminant has norm at most X.  Every permutation g in G has an index, which is just n – the number of orbits of g.  So the permutations of index 1 are the transpositions, those of index 2 are the three-cycles and the double-flips, etc.  We denote by a(G) the reciprocal of the minimal index of any element of G.  In particular, a(G) is at most 1, and is equal to 1 if and only if G contains a transposition.

(Wait, doesn’t a transitive subgroup of S_n with a transposition have to be the whole group?  No, that’s only for primitive permutation groups.  D_4 is a thing!)

Malle’s conjecture says that, for every \epsilon > 0, there are constants c,c_\epsilon such that

c X^{a(G)} < N(G,K,X) < c_\epsilon X^{a(G)+\epsilon}

We don’t know much about this.  It’s easy for G = S_2.  A theorem of Davenport-Heilbronn (K=Q) and Datskovsky-Wright (general case) proves it for G = S_3.  Results of Bhargava handle S_4 and S_5, Wright proved it for abelian G.  I kind of think this new theorem of Alex Smith implies for K=Q and every dihedral G of 2-power order?  Anyway:  we don’t know much.  S_6?  No idea.  The best upper bounds for general n are still the ones I proved with Venkatesh a long time ago, and are very much weaker than what Malle predicts.

Malle’s conjecture fans will point out that this is only the weak form of Malle’s conjecture; the strong form doesn’t settle for an unspecified X^\epsilon, but specifies an asymptotic X^a (log X)^b.   This conjecture has the slight defect that it’s wrong sometimes; my student Seyfi Turkelli wrote a nice paper which I think resolves this problem, but the revised version of the conjecture is a bit messy to state.

Anyway, here’s the new theorem:

Theorem (E-Tran-Westerland):  Let G be a transitive subgroup of S_n.  Then for all q sufficiently large relative to G, there is a constant c_\epsilon such that

N(G,\mathbf{F}_q(t),X) < c_\epsilon X^{a(G)+\epsilon}

for all X>0.

In other words:

The upper bound in the weak Malle conjecture is true for rational function fields.

A few comments.

  1.  We are still trying to fix the mistake in our 2012 paper about stable cohomology of Hurwitz spaces.  Craig and I discussed what seemed like a promising strategy for this in the summer of 2015.  It didn’t work.  That result is still unproved.  But the strategy developed into this paper, which proves a different and in some respects stronger theorem!  So … keep trying to fix your mistakes, I guess?  There might be payoffs you don’t expect.
  2. We can actually bound that X^\epsilon is actually a power of log, but not the one predicted by Malle.
  3. Is there any chance of getting the strong Malle conjecture?  No, and I’ll explain why.  Consider the case G=S_4.  Then a(G) = 1, and in this case the strong Malle’s conjecture predicts N(S_4,K,X) is on order X, not just X^{1+eps}.   But our method doesn’t really distinguish between quartic fields and other kinds of quartic etale algebras.  So it’s going to count all algebras L_1 x L_2, where L_1 and L_2 are quadratic fields with discriminants X_1 and X_2 respectively, with X_1 X_2 < X.  We already know there’s approximately one quadratic field per discriminant, on average, so the number of such algebras is about the number of pairs (X_1, X_2) with X_1 X_2 < X, which is about X log X.  So there’s no way around it:  our method is only going to touch weak Malle.  Note, by the way, that for quartic extensions, the strong Malle conjecture was proved by Bhargava, and he observes the same phenomenon:

    …inherent in the zeta function is a sum over all etale extensions” of Q, including the “reducible” extensions that correspond to direct sums of quadratic extensions. These reducible quartic extensions far outnumber the irreducible ones; indeed, the number of reducible quartic extensions of absolute discriminant at most X is asymptotic to X log X, while we show that the number of quartic field extensions of absolute discriminant at most X is only O(X).

  4.  I think there is, on the other hand, a chance of getting rid of the “q sufficiently large relative to G” condition and proving something for a fixed F_q(t) and all finite groups G.


OK, so how did we prove this?

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