## 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?

## Bhargava and Satriano on Galois closures of rings

Manjul Bhargava and Matt Satriano (starting a postdoc at Michigan this fall) posted a nice paper on the arXIv, “On a notion of “Galois closure” for extensions of rings.” The motivation for this work (I’m guessing) comes from Bhargava’s work on parametrizations of number fields.  Bhargava needs to generalize many classical objects of algebraic number theory from the setting of fields to rings.  For instance, the sextic resolvent of a degree 5 polynomial f(x) (developed by Lagrange, Malfatti, and Vandermonde in the late 18th century) is a degree 6 polynomial g(x) which has a rational root if and only if the equation f(x)=0 is solvable in radicals.

In modern Galois-theoretic language, we would describe the sextic resolvent as follows.  If K/Q is the Galois closure of the field generated by a root of f, then the Galois group Gal(K/Q) acts on the 5 roots of f, and is thus identified with a subgroup of S_5.  On the other hand, S_5 has an index-6 subgroup H, the normalizer of a 5-cycle.  So the action of Gal(K/Q) on S_5 / H gives rise to an extension of Q of degree 6 (maybe a field, maybe a product of fields) and this is Q[x] / g(x).

So far, so good.  But Bhargava needs to define a sextic resolvent which is a rank-6 free Z-module, not a 6-dimensional Q-vector space.  The sticking point is the notion of “Galois closure.”  What is the Galois closure of a rank-5 algebra over Z?  Or, for that matter, over a general commutative ring?

Bhargava gets around this question in his paper on quintic fields by using a concrete construction particular to the case of quintics.  But in the new paper, he and Satriano propose a very nice (“very nice” means “functorial”) completely general construction of a “Galois closure” G(A/B) for any extension of rings A/B such that A is a locally free B-module of rank n.  G(A/B) is a locally free B-module, endowed with an S_n-action, as you might want, and it agrees with the usual definition when A/B is an extension of fields.

But there are surprises — for instance, the Galois closure of the rank 4 algebra C[x,y,z]/(x,y,z)^2 over C is 32-dimensional!  In fact, the authors show that there is no definition of Galois closure which is functorial and for which G(A/B) always has the “expected” rank n! over B.  This might explain why no one has written down this definition before, and I think it is what gives the paper a sort of offbeat charm.  It illustrates a useful point:  you’ve got to know when it’s time to mulch an axiom.

## Math And: Arielle Saiber on Italian poetry and Italian algebra, Friday, Oct 23 at 4pm Something to do tomorrow (besides eating the Beef n Brew slice): the Math And… seminar is very pleased to welcome Arielle Saiber from Bowdoin for our Fall 2009 lecture.  Arielle is an Italianist of very broad interests, with academic papers on Italian literature, the early history of algebra and geometry, Dali’s illustrations for Dante, and the polyvalent discourse of electronic music.  Tomorrow there will only be time to unite the first two.

23 Oct 2009, 4pm, Van Vleck B239: Arielle Saiber (Bowdoin, Italian)

Title “Nicollo Tartaglia’s Poetic Solution to the Cubic Equation.”

Niccolo Tartaglia’s (1449-1557) solution to solving cubic equations, which renowned mathematician and physician Girolamo Cardano wanted but Tartaglia resisted, led to one of the first intellectual property cases in Western history. Eventually, Tartaglia agreed to give Cardano what he so desired, but only if the latter promised he would not publish it. Cardano promised, and Tartaglia sent him the solution. Wasting little time, however, Cardano published the solution (along with a ‘general’ solution that he himself developed). Tartaglia was, not surprisingly, furious and began a vicious battle with Cardano’s assistant, Ludovico Ferrari (Cardano refused to engage Tartaglia directly). But vitriolic polemics aside, there is something else rather curious about this ordeal: the solution Tartaglia gave Cardano was encrypted in a poem. This talk looks at the motives behind his “poetic solution” and what it says about the close relationship between ‘poeisis’ and ‘mathesis’ in this period of mathematics’ history.