Statistics of Number Fields and Function Fields

We discuss some problems of arithmetic distribution, including conjectures of Cohen-Lenstra, Malle, and Bhargava; we explain how such conjectures can be heuristically understood for function fields over finite fields, and discuss a general approach to their proof in the function field context based on the topology of Hurwitz spaces. This approach also suggests that the Schur multiplier plays a role in such questions over number fields. 1. Arithmetic Counting Problems We begin with a concrete example, which has been well-understood for many years. Let SX denote the set of squarefree integers in [0, X] that are congruent to 1 modulo 4; let CX denote the set of isomorphism classes of cubic field extensions K/Q whose discriminant belongs to SX . Davenport and Heilbronn proved [6] that |CX | |SX | −→ 1 6 , as X →∞. (1) Our goal is to understand why limits like that of (1) should exist, why they should be rational numbers, and what the rational numbers represent. More precisely, we will study several variants on (1) – replacing cubic fields by extensions with prescribed Galois group, and “squarefree discriminant” by other forms of prescribed ramification. We make a heuristic argument as to what the corresponding limits should be when Q is replaced by the function field of a curve over a finite field, and lay out a program for a proof in certain cases. This program has been partially implemented by us in certain settings, leading to a weak form of the Cohen–Lenstra heuristics (see §4.2) over a rational function field. In the number field case we have no new theorems; however, the study ∗Akshay Venkatesh, Stanford University. E-mail: [email protected]. †Jordan Ellenberg, University of Wisconsin. E-mail: [email protected]. 2 Akshay Venkatesh and Jordan S. Ellenberg of the function field case suggests interesting refinements of known heuristics, related to the size of Schur multipliers. Let us describe – briefly and approximately – how the 16 makes an appearance over a function field. Let k be a finite field, and k̄ an algebraic closure of k; we consider cubic extensions L of k(t) with squarefree discriminant and totally split at ∞. By a marking of L we shall mean an ordering of the three places above ∞. “Marked” cubic extensions can be descended: they are identified with fixed points of a Frobenius acting on marked cubic extensions of k̄(t). Recall that the average number of fixed points of a random permutation on a finite set is 1; thus, if the Frobenius behaves like a random permutation, we expect there to be on average one marked cover per squarefree discriminant. Since there are six markings for each cubic field that is totally split at ∞, we recover 1 6 . The rest of this paper will discuss methods for trying to make this heuristic into a proof, and how it suggests corrections to our view of number fields. In the function field case, results such as (1) are related to the group-theoretic structure of étale π1; we may speculate that results such as (1) are reflections of some (as yet, not understood) group-theoretic features of the absolute Galois group of Q. 1.1. Context. There has been a great deal of work on the topics discussed here. We note in particular that related topics have been discussed [1, 3, 7] in the last three ICMs. Indeed, [1] contains an overview of Bhargava’s results for quartic and quintic fields, and [3] discusses both theoretical and numerical evidence for conjectures of the type described in the present paper. Our point of view is influenced very much by the study of the function field case; in turn, our study of that case was influenced by both Cohen and Lenstra’s work and the more recent paper [8] of Dunfield and Thurston on finite covers of hyperbolic 3-manifolds. The present paper has three sections; although related, they are also to a large extent independent, and can be considered as “variations on the theme of (1).” – §2 discusses the conjectures of Bhargava-Malle about distribution of number fields, generalizing (1). We also discuss the role that Schur multipliers may play in formulating sharp versions of such conjectures (§2.4). The reader may wish to first look at Section 3.4, which provides the geometric motivation guiding the computations in Sections 2.4 and 2.5. – §3 discusses the function field setting and its connection with the geometry of Hurwitz spaces; in particular, how purely topological results on the stable homology of Hurwitz spaces would imply function field versions of Bhargava-Malle conjectures. – §4 discusses the special case of the Cohen–Lenstra heuristics, and our proof (with Westerland) of a weak version in the function field setting. Statistics of Number Fields and Function Fields 3 This proof suggests more general connections between analytic number theory and stable topology. 1.2. Notation. By a G-extension algebra (resp. field) of a number field K, we shall mean a conjugacy class of homomorphisms (resp. surjective homomorphisms) from the Galois group GK := Gal(K/K) to G. In other words, G-extension fields are in correspondence with isomorphism classes of pairs (L ⊃ K,G ∼ → Gal(L/K)), where an isomorphism of pairs (L, f) and (L′, f ′) is simply a K-isomorphism φ : L → L′ which commutes with the induced G-actions. A pair (G, c) of a group G and a conjugacy class c ⊂ G will be called admissible – for short, we say that c is an admissible conjugacy class – if 1. c is a rational conjugacy class, i.e., g ∈ c =⇒ g ∈ c whenever n is prime to the order of g; 2. c generates the group G. Given a tamely ramified G-extension and an admissible conjugacy class, we say that all ramification is of type c if the image of every inertia group is either trivial or a cyclic subgroup generated by some g ∈ c. Acknowledgements. We thank Craig Westerland, our collaborator on the work described here, for many years of advice and ideas about the topological side of the subject. We have also greatly benefited from conversations with Manjul Bhargava, Nigel Boston, Ralph Cohen, Henri Cohen, David Roberts, and Melanie Wood.