My Ph.D. student Seyfi Turkelli recently posted a really nice paper, “Connected components of Hurwitz schemes and Malle’s conjecture,” to the arXiv. It’s a beautiful example of the “hidden geometry” behind questions about arithmetic distributions, so I thought I’d say a little about it here.

The story begins with the old conjecture, sometimes attributed to Linnik, that the number of degree-n extensions of Q of discriminant at most X grows linearly with X, as X grows with n held constant. When n=2, this is easy; when n = 3, it is a theorem of Davenport and Heilbronn; when n=4 or 5, it is recent work of Bhargava; when n is at least 6, we have no idea.

Having no idea is, of course, no barrier to generalization. Here’s a more refined version of the conjecture, due to Gunter Malle. Let K be a number field, let G be a finite subgroup of S_n, and let N_{K,G}(X) be the number of extensions L/K of degree n whose discriminant has norm at most K, and whose Galois closure has Galois group G. Then there exists a constant c_{K,G} such that

**Conjecture**: N_{K,G}(X) ~ c_{K,G} X^a(G) (log X)^(b(K,G))

where a and b are constants explicitly described by Malle. (Malle doesn’t make a guess as to the value of c_{K,G} — that’s a more refined statement still, which I hope to blog about later…)

Akshay Venkatesh and I wrote a paper (“Counting extensions of function fields…”) in which we gave a heuristic argument for Malle’s conjecture over K = F_q(t). In that case, N_{K,G}(X) is the number of points on a certain *Hurwitz space*, a moduli space of finite covers of the projective line. We were able to control the dimensions and the number of irreducible components of these spaces, using in a crucial way an old theorem of Conway, Parker, Fried, and Volklein. The heuristic part arrives when you throw in the 100% shruggy guess that an irreducible variety of dimension d over F_q has about q^d points. When you apply this heuristic to the Hurwitz spaces, you get Malle’s conjecture on the nose.

So we were a little taken aback a couple of years later when Jurgen Kluners produced counterexamples to Malle’s theorem! We quickly figured out what was going on. There wasn’t anything wrong with our theorem; just our analogy. Our Hurwitz spaces were counting *geometrically connected* covers of the projective line. But a cover Y -> P^1/F_q which is connected, but not geometrically connected, provides a perfectly good field extension of F_q(t). If we’re trying to imitate the number field question, we’d better count those too. It had never occurred to us that they might outnumber the geometrically connected covers — but that’s just what happens in Kluners’ examples.

What Turkelli does in his new paper is to work out the dimensions and components for certain *twisted* Hurwitz spaces which parametrize the connected but not geometrically connected covers of P^1. This is a really subtle thing to get right — you can’t rely on your geometric intuition, because the phenomenon you’re trying to keep track of doesn’t exist over an algebraically closed field! But Turkelli nails it down — and as a consequence, he gets a new version of Malle’s conjecture, which is compatible with Kluners’ examples, and which I think is really the right statement. Which is not to say I know whether it’s true! But if it’s not, it’s at least the correct false guess given our present state of knowledge.

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