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 , there are constants such that

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 , but specifies an asymptotic . 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 such that

for all X>0.

In other words:

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

A few comments.

- 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.
- We can actually bound that is actually a power of log, but not the one predicted by Malle.
- 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).

- 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?