On class groups of random number fields

The main aim of this paper is to disprove the Cohen–Lenstra–Martinet heuristics in two different ways and to offer possible corrections. We also recast the heuristics in terms of Arakelov class groups, giving an explanation for the probability weights appearing in the general form of the heuristics. We conclude by proposing a rigorously formulated Cohen–Lenstra–Martinet conjecture.


Introduction
The Cohen-Lenstra-Martinet heuristics [4,6] make predictions on the distribution of class groups of "random" algebraic number fields. In the present paper, we disprove the predictions in two different ways, and propose possible adjustments. In addition, we show that the heuristics can be equivalently formulated in terms of Arakelov class groups of number fields. This formulation has the merit of conforming to the general expectation that a random mathematical object is isomorphic to a given object A with a probability that is inversely proportional to # Aut A. We end by offering two rigorously formulated Cohen-Lenstra-Martinet conjectures that appear to be consistent with everything we know. In particular, we give two possible precise definitions of the notion of a "reasonable function" occurring in [4,6].
The class group Cl F of a number field F is in a natural way a module over Aut F . Our first disproof of the heuristics relies on the following theorem, which places restrictions on the possible module structure. Recall that for a ring R, the Grothendieck group G(R) of the category of finitely generated R-modules has one generator [L] for every finitely generated R-module L, and one defining relation [L] + [N ] = [M ] for every short exact sequence 0 → L → M → N → 0 of finitely generated R-modules. If S is a set of prime numbers, we write Z (S) = {a/b : a, b ∈ Z, b ∈ p∈S∪{0} pZ}, which is a subring of Q. If F is a number field, let µ F denote the group of roots of unity in F . Greither had proven in [8,Theorem 5.5] a stronger version of Theorem 1.1 under some additional hypotheses. We will prove Theorem 1.1 in Section 4 as a consequence of the Iwasawa Main Conjecture for abelian fields, as proven by Mazur-Wiles in [14]. We will then show that the theorem contradicts the Cohen-Martinet heuristics as follows. Suppose that G is cyclic of order 58, and S consists of all prime numbers not dividing 58. Then G(T − ) tors is finite non-trivial, and, as we will argue in Section 4, the Cohen-Martinet heuristics predict that, when F ranges over all imaginary fields with this G, the class Cl F ] will turn out to be trivial for all but one F , so the prediction is wrong in this case. In the Cohen-Lenstra-Martinet conjecture that we propose below, we will remove this obstruction by requiring the set S to be finite, in which case one has G(T − ) tors = 0.
Our second disproof of the Cohen-Martinet heuristics is of a different order. For a positive real number x, let C(x) be the set of cyclic quartic fields with discriminant at most x inside a fixed algebraic closure of Q, and let C ′ (x) ⊂ C(x) be the subset of those for which the class number of the quadratic subfield is not divisible by 3. Then, as we will explain in Section 6, the Cohen-Martinet heuristics predict that the limit lim x→∞ #C ′ (x) #C(x) exists and that lim x→∞ #C ′ (x) #C(x) ≈ 0.8402, where the notation a ≈ b means that a rounds to b with the given precision. In Section 6, we will prove the following result, disproving this prediction. Enumerating cyclic quartic fields in the order of non-decreasing discriminant has the undesirable feature that every quadratic field that occurs at all as a subfield does so with positive probability. It is this observation that allows us to prove Theorem 1.2. Accordingly, we propose to use an order of enumeration that, by work of Matchett Wood [13], does not exhibit this behaviour, and is not expected to do so in the generality of our conjecture. Enumerating number fields by discriminant has also been observed to pose problems in the context of other questions in arithmetic statistics, which are not obviously related to the Cohen-Martinet heuristics, see e.g. [13,1].
Let us now discuss the shift of perspective towards Arakelov class groups. Let F be a number field. For the definition of the Arakelov class group Pic 0 F of F , we refer to [18]. It may be compactly described as the cokernel of the natural map ( J F ) → ( J F /F × ), where J F denotes the idèle group of F (see [3]) and (X) denotes the maximal compact subgroup of X; in particular, Pic 0 F is a compact abelian group. We denote the Pontryagin dual of Pic 0 F by Ar F . It is an immediate consequence of [18,Proposition 2.2] that Ar F is a finitely generated discrete abelian group that fits in a short exact sequence of Aut F -modules where O F denotes the ring of integers of F . Thus, knowing the torsion subgroup of Ar F is equivalent to knowing Cl F , and knowing its torsion-free quotient amounts to knowing O × F modulo roots of unity. The Arakelov class group of a number field is often better behaved than either the class group or the integral unit group. As an example of this phenomenon, we mention the following analogue of Theorem 1.1 for real abelian fields, which we will prove in Section 5, also relying on the results of Mazur-Wiles [14]. As we will explain in Section 5, this theorem expresses that the class of T ⊗ Z[G] Ar F in G(T ) is "as trivial as it can be", given the Q[G]-module structure of Q[G] ⊗ Z[G] Ar F . There is no reason to believe that an analogous result holds for either of the other two terms in the exact sequence (1.3).
We also reinterpret Theorems 1.1 and 1.4 in terms of the so-called oriented Arakelov class group, a notion that was introduced by Schoof in [18] and of which we recall the definition in Section 5. This reinterpretation prompts us to ask the following question. Question 1.5. Let K be an algebraic number field, let d be its degree over Q, let F/K be a finite Galois extension, let G be its Galois group, let S be a set of prime numbers not dividing 2 · #G, let T = Z (S) [G], and let Ar F denote the Pontryagin dual of the oriented Arakelov class group of F . Is the In Theorem 5.4 we answer the question in the affirmative when K = Q and G is abelian.
Above we mentioned the principle that, if a mathematical object is "randomly" drawn, a given object appears with a probability that is inversely proportional to the order of its automorphism group. In the context of Arakelov class groups, however, the relevant automorphism groups are typically of infinite order. In [2], we overcame this obstacle to applying the principle by means of an algebraic theory, the consequences of which we now explain.
Let G be a finite group, and let A be a quotient of the group ring Q[G] by some two-sided ideal. If p is a prime number and S = {p}, then we write Z (p) for Z (S) . We say that a prime number p is good for A if there is a direct product decomposition Z (p) [G] = J × J ′ , where J is a maximal Z (p) -order in A, and the quotient map For example, all prime numbers p not dividing #G are good for all quotients of Q[G]. Let S be a set of prime numbers that are good for A, and let R denote the image of Z (S) [G] in A. Let M be a set of finite R-modules with the property that for every finite R-module M ′ there is a unique M ∈ M such that M ∼ = M ′ , and let P be a set of finitely generated projective R-modules such that for every finitely generated projective R-module P ′ there is a unique P ∈ P such that P ∼ = P ′ . Note that M and P are countable sets.
Assume for the rest of the introduction that S is finite. As we will show in Section 3, it follows from our hypotheses that for every finitely generated A-module V , there is a unique P V ∈ P such that A ⊗ R P V ∼ = A V ; and that moreover for every finitely generated R-module M satisfying Let V be a finitely generated A-module. In Section 3, we will deduce from [2] that there is a unique discrete probability measure P on M V with the property that for any isomorphism L ⊕ E ∼ = M of R-modules, where L and M are in M V , and E is finite, we have where the inclusion Aut L ⊂ Aut M is the obvious one. This condition expresses that one can think of P(M ) as being proportional to 1/# Aut M .
We will now formulate an incomplete version of our conjecture on distributions of Arakelov class groups, leaving the discussion of the crucial missing detail, the notion of a "reasonable" function, to Section 7. If f is a complex valued function on M V , then we define the expected value of f to be the sum E(f ) = M ∈M V (f (M ) · P(M )) if the sum converges absolutely.
Let G, A, S, R, and V be as above, and assume that g∈G g = 0 in A. Let K be a number field, letK be an algebraic closure of K, and let F be the set of all pairs (F, ι), where F ⊂K is a Galois extension of K that contains no primitive p-th root of unity for any prime p ∈ S, and ι is an isomorphism between the Galois group of F/K and G that induces an For all (F, ι) ∈ F, we will view Ar F as a G-module via the isomorphism ι. Let (F, ι) ∈ F be arbitrary. It follows from the exact sequence (1.3) that we have an isomorphism A ⊗ Z[G] Ar F ∼ = Hom(V, Q) of A-modules. Moreover, every finitely generated Q[G]-module is isomorphic to its Q-linear dual (see e.g. [7, §10D]), so we have an isomorphism Hom(V, If F/K is a finite extension, let c F/K be the ideal norm of the product of the prime ideals of O K that ramify in F/K. For a positive real number B, let F c≤B = {(F, ι) ∈ F : c F/K ≤ B}. If M is a finitely generated R-module satisfying A ⊗ R M ∼ = A V , and f is a function defined on M V , then we write f (M ) for the value of f on the unique element of M V that is isomorphic to M . Conjecture 1.6. Let f be a "reasonable" complex valued function on M V . Then the limit #F c≤B exists, and is equal to E(f ).
The notion of a "reasonable" function already appears in the original conjectures of Cohen-Lenstra [4] and Cohen-Martinet [6], but has, to our knowledge, never been made precise. We will offer two possible definitions of this notion in Section 7.
Of course, the minimal requirement that a function f has to fulfil in order to satisfy the conclusion of Conjecture 1.6 is that the expected value E(f ) should exist. It may be tempting to conjecture that, at least for R ≥0valued functions f , this minimal condition is in fact sufficient. However, the following result, which we will prove as Theorem 7.1, using a construction communicated to us by Bjorn Poonen, shows that the conjecture is likely false in that generality. Theorem 1.7. Let X be a countably infinite set, and let p be a discrete probability measure on X. For all x ∈ X, abbreviate p({x}) to p(x) and assume that p(x) > 0. Let B be the subset of X Z ≥1 consisting of those sequences (y i ) i∈Z ≥1 ∈ X Z ≥1 for which there exists a function f : Then the measure of B with respect to the product measure induced by p on X Z ≥1 is equal to 1. Theorem 1.7 suggests that if the sequence of dual Arakelov class groups really were a random sequence with probability measure P, then with probability 1 there would be a function f for which E(f ) exists but the conclusion of Conjecture 1.6 is violated.
In order to allow the reader to compare our conjecture with the Cohen-Lenstra-Martinet heuristics, we will prove in Section 3 that the probability distribution P V on M that P induces on the Z (S) -torsion submodules of M ∈ M V satisfies, for all L 0 , M 0 ∈ M, The distribution P V is, in fact, the probability distribution that is used in the original Cohen-Lenstra-Martinet heuristics, a version of which we will give in Section 2.
done at the MPIM in Bonn, and most of it was done while the first author was at Warwick University. We would like to thank these institutions for their hospitality and financial support.

The Cohen-Lenstra-Martinet heuristics
In this section, we give a version of the heuristics of Cohen-Lenstra and Cohen-Martinet. Our formulation differs from the original one in several ways, as we will point out, but does not yet incorporate the corrections that will be necessary in light of the results of Sections 4 and 6.
Let G be a finite group, let A be a quotient of the group ring Q[G] by some two-sided ideal that contains g∈G g, let S be a set of prime numbers that are good for A, let R be the image of Z (S) [G] in A under the quotient map, let P be a finitely generated projective R-module, and let V denote the A-module A ⊗ R P . Let M be as in the introduction. If B = (B A ′ ) A ′ is a sequence of real numbers indexed by the simple quotients A ′ of A, then let M ≤B be the set of all M ∈ M such that for every simple quotient If f is a function on M, and M is a finite R-module, then we will write f (M ) for the value of f on the unique element of M that is isomorphic to M . For a finite module M , define w P (M ) = 1 # Hom(P,M )·# Aut M . Let K be a number field, letK be an algebraic closure of K, and let F be the set of all pairs (F, ι), where F ⊂K is a Galois extension of K that contains no primitive p-th root of unity for any prime p ∈ S, and ι is an isomorphism between the Galois group of F/K and G that induces is in F, then the class group Cl F is naturally a G-module via the isomorphism ι. Let ∆ F/K denote the ideal norm of the relative discriminant of F/K. For a positive real number B, let F ∆≤B = {(F, ι) ∈ F : ∆ F/K ≤ B}.
Heuristic 2.1 (Cohen-Lenstra-Martinet heuristics). With the above notation, assume that F is infinite, and let f be a "reasonable" C-valued function on M. Then: exists;
The notion of a "reasonable" function was left undefined in [4] and [6], and has, to our knowledge, never been made precise.
Let us briefly highlight some of the differences between Heuristic 2.1 and [6, Hypothèse 6.6].
While Cohen-Martinet assume that the families of fields under consideration are non-empty, we assume in Heuristic 2.1 that F is infinite. If F was finite, then Heuristic 2.1 would almost certainly be false for any reasonable interpretation of the word "reasonable", so we avoid dependencies on some form of the inverse Galois problem.
The original heuristics did not include the condition that the fields F should not contain a primitive p-th root of unity for any p ∈ S. However, work by Malle [12] suggests that if that condition was omitted, then for those primes p that do divide the order of the group of roots of unity of F for a positive proportion of all (F, ι) ∈ F, the probability measure governing the p-primary parts of the class groups would likely need to be modified.
Considering pairs (F, ι) consisting of a number field and an isomorphism between its Galois group and G, as in Heuristic 2.1, is one way of making precise the original formulation of Cohen-Martinet, who speak of the family of extensions of K "with Galois group G". In the above formulation, some concrete conjectures of [5] become trivially true. An example of this is [5, (2)(a)], where it is conjectured that if C 3 is a cyclic group of order 3, then among Galois number fields with Galois group isomorphic to C 3 and with class number 7, each of the two isomorphism classes of non-trivial C 3 -modules of order 7 appears with probability 50%. We do not know how to make the notion of a family of fields "with Galois group G" precise in a natural way that would render this and similar examples well-defined, but not trivial. In There are some curious functions that may be considered reasonable, but for which the limit (2.3) does not exist when S is too large. Suppose, for example, that S contains almost all prime numbers, and that R is the product of two rings T and is fixed and B C tends to ∞, then that limit is 1. From this observation it can be deduced that the limit in (2.3) does not exist. Such examples can be realised in the context of number fields: let C 2 be a cyclic group of order 2 and S 3 the symmetric group on a set of 3 elements, suppose that F/Q is Galois with Galois group isomorphic to C 2 × S 3 such that the inertia groups at ∞ are subgroups of S 3 of order 2, and let S contain almost all prime numbers. Then F contains two imaginary quadratic subfields, one that is contained in a subextension that is Galois over Q with Galois group isomorphic to S 3 , and one that is not, and the question of how often the order of the S-class group of the former is greater than that of the latter cannot be answered by Heuristic 2.1. We leave it to the reader to check that this example can indeed be realised in the framework of Heuristic 2.1 with a judicious choice of A and V .
If instead S is finite, then all the relevant sums converge absolutely, and the limit (2.3) is well-defined. This adds to our reasons for demanding in Conjecture 1.6 that S be finite.

Commensurability of automorphism groups
In this section, we recall the formalism of commensurability from [2], and deduce the existence of the probability measure P as described in the introduction.
A group isogeny is a group homomorphism f : H → G with # ker f < ∞ and (G : f H) < ∞, and its index i(f ) is defined to be (G : f H)/# ker f . For a ring R, an R-module isogeny is an R-module homomorphism that is an isogeny as a map of additive groups. A ring isogeny is a ring homomorphism that is an isogeny as a map of additive groups. The index of an isogeny of one of the latter two types is defined as the index of the induced group isogeny on the additive groups.
If X, Y are objects of a category C, then a correspondence from X to Y in C is a triple c = (W, f, g), where W is an object of C and f : W → X and g : W → Y are morphisms in C; we will often write c : X ⇋ Y to indicate a correspondence. A group commensurability is a correspondence c = (W, f, g) in the category of groups for which both f and g are isogenies. For a ring R, we define an R-module commensurability to be a correspondence of R-modules that is a commensurability of additive groups, and a ring commensurability is defined analogously. If c = (W, f, g) is a commensurability of groups, or of rings, or of modules over a ring, then its index is defined as i(c) = i(g)/i(f ).
Let R be a ring, and let c = (N, f, g) : L ⇋ M be a correspondence of R-modules. We define the endomorphism ring End c of c to be the subring There are natural ring homomorphisms End c → End L and End c → End M sending (λ, ν, µ) to λ and µ, respectively; we shall write e(c) : End L ⇋ End M for the ring correspondence consisting of End c and those two ring homomorphisms. Similarly, we define the automorphism group Aut c of c to be the group (End c) × , and we write a(c) : Aut L ⇋ Aut M for the group correspondence consisting of Aut c and the natural maps Aut c → Aut L, Aut c → Aut M .
The following result is a special case of [2, Theorem 1.2].
in A, and let L, M be finitely generated R-modules. Then: Proof. By Theorem 3.1(a), there exist commensurabilities L ⇋ M , P ⇋ L, and P ⇋ M . We will first compute ia(P, L) and ia(P, M ). The split exact sequence 0 → L 0 → L π → P → 0, where π is the natural projection map, induces a surjective map Aut L → Aut L 0 × Aut P, whose kernel is easily seen to be canonically isomorphic to Hom(P, L 0 ). It follows that if c is the commensurability (L, π, id) : P ⇋ L, then the map Aut c → Aut L is an isomorphism, while the map Aut c → Aut P is onto, with kernel of cardinality # Hom(P, L 0 ) · # Aut L 0 . Hence ia(P, L) = i(a(c)) = # Hom(P, L 0 )·# Aut L 0 , and analogously for ia(P, M ). By Lemma 3.3, we therefore have that as claimed.
For the rest of the section, assume that S is a finite set of prime numbers that are good for A, and let M and P be sets of finitely generated R-modules as in the introduction.
Proof. Let T be any finitely generated subgroup of V such that QT = V . Then P = RT is a finitely generated R-submodule of V such that A ⊗ R P = V . By [16,Corollary (11.2)], the ring R is a maximal Z (S) -order in A. It follows from [16, Theorems (18.1) and (2.44)] that P is a projective Rmodule, and is isomorphic to a unique element P V of P.
Let M be a finitely generated R-module such that A ⊗ R M ∼ = A V , let M tors be the R-submodule of M consisting of Z (S) -torsion elements, and letM = M/M tors be the Z (S) -torsion free quotient. It follows from [16,Theorem (18.10) Recall from the introduction that if V is a finitely generated A-module, and We now state and prove the main result of the section. Proposition 3.6. Under the assumptions of Notation 3.2, suppose that S is a finite set of prime numbers that are good for A, let V be a finitely generated A-module, and let P V ∈ P be such that A ⊗ R P V ∼ = V . Then: (a) there exists a unique discrete probability distribution P V on M such that for all L 0 , M 0 ∈ M we have By [6, Théorème 3.6], the sum L 0 ∈M w(L 0 ) converges, to α, say, so that we may define the probability distribution P V on M by P V (L 0 ) = w(L 0 )/α for L 0 ∈ M. It satisfies the conclusion of part (a), and is clearly the unique such distribution. This proves part (a).
We now prove part (b). By combining the convergence of L 0 ∈M w(L 0 ) with Proposition 3.4, we see that for all M ∈ M V , the sum L∈M V ia(L, M ) also converges, to β M , say, so that we may define a probability distribution Thence it immediately follows that P satisfies the conclusion of part (b), and it is clearly the unique such probability distribution.

Class groups of imaginary abelian fields
In the present section, we prove Theorem 1.1, and use it to give a disproof of Heuristic 2.1. We begin by establishing some notation for the section and recalling some well-known facts that will also be useful in the next section.
Generalities on group rings. LetQ be an algebraic closure of Q. Let G be a finite abelian group, with dualĜ = Hom(G, of which the image is the cyclotomic field Q(χ(G)), and the natural map is an isomorphism of Q-algebras. Let S be a set of prime numbers not dividing #G, and write , which is a Dedekind domain. We have a ring isomorphism T ∼ = χ∈Ĝ/∼ χ(T ), so each T -module M decomposes as a direct sum χ∈Ĝ/∼ (χ(T ) ⊗ T M ), which leads to a group isomorphism G(T ) ∼ = χ∈Ĝ/∼ G(χ(T )), where we recall from the introduction that G denotes the Grothendieck group.
Since for each χ ∈Ĝ the ring is the class group of the Z (S) -order χ(T ), and in particular is finite. Explicitly, the projection map G(χ(T )) → Z is defined by sending the class of a finitely generated Roots of unity. Let F/Q be an imaginary abelian field with Galois group G, and let c ∈ G be the automorphism of F given by complex conjugation. We writeĜ − = {χ ∈Ĝ : χ(c) = −1}, and T − = T /(1 + c); the latter ring may be identified with χ∈Ĝ − /∼ χ(T ), and one has For each m ∈ Z >0 , we denote by ζ m a primitive m-th root of unity in some algebraic closure of F . Let U be the set of prime numbers q ∈ S for which F contains ζ q . Recall from the introduction that µ F denotes the group of roots of unity in F .
Proof. Since µ F is finite cyclic, the group Z (S) ⊗ Z µ F is cyclic of order equal to the largest divisor of #µ F that is a product of primes in S. If q is a prime number for which q 2 divides #µ F , then q divides [F : Q] = #G, and therefore q ∈ S. This implies that Z (S) ⊗ Z µ F is cyclic of order q∈U q. It is also a Z (S) [G]-module on which c acts as −1, so it equals For each q ∈ U , denote by ϕ q : T − → End ζ q ∼ = Z/qZ the ring homomorphism that describes the T − -module structure of ζ q .
Proposition 4.3. For each q ∈ U there is an element χ q ∈Ĝ − , unique up to ∼, such that ϕ q factors as T − → χ q (T ) → Z/qZ, and it is characterised by the subfield F ker χq ⊂ F being equal to Q(ζ q ). Also, if p q denotes the kernel of χ q (T ) → Z/qZ, then the image of the element under the isomorphism (4.1) equals the image of (χ q (T )/p q ) q∈U under the natural inclusion q∈U G(χ q (T )) ⊂ χ∈Ĝ − /∼ G(χ(T )).
Proof. Since Z/qZ is a field, the map ϕ q factors through exactly one of the components in the decomposition T − = χ∈Ĝ − /∼ χ(T ), say through χ q (T ). By the irreducibility of the q-th cyclotomic polynomial, the induced map G → χ q (G) → (Z/qZ) × is surjective, and since q does not divide #G, the map χ q (G) → (Z/qZ) × is injective, so the map χ q (G) → (Z/qZ) × is an isomorphism. This shows that ker χ q = ker(G → (Z/qZ) × ), so we have indeed F ker χq = Q(ζ q ). In particular, the order of χ q equals q − 1, so χ q = χ q ′ for distinct q, q ′ ∈ U . We have an isomorphism of T − -modules T − ⊗ Z[G] µ F ∼ = q∈U χ q (T )/p q , and this implies the last assertion.  Example 4.4. The prime ideal p q of χ q (T ) occurring in Proposition 4.3 equals m q,ωq , where ω q : G → Z × q is the unique group homomorphism for which the induced map G → (Z/qZ) × describes the G-module structure of ζ q . The character ω q ∈Ĝ is called the Teichmüller character at q.
Bernoulli numbers. Let χ ∈Ĝ − , and let f (χ) ∈ Z >0 be minimal with F ker χ ⊂ Q(ζ f (χ) ). For each t ∈ Z that is coprime to f (χ), denote by η t the restriction to F ker χ of the automorphism of Q(ζ f (χ) ) that sends ζ f (χ) to ζ t f (χ) ; note that η t belongs to the Galois group Gal(F ker χ /Q), which may be identified with G/ ker χ and with χ(G), and indeed we shall view η t as an element of the cyclotomic field Q(χ(G)). We define the Bernoulli number which is also an element of the cyclotomic field Q(χ(G)). For each (p, ψ) ∈ S × [χ], the character ψ induces a field embedding Q(χ(G)) →Q p , which we simply denote by ψ. The following result relates the image ψ(β(χ)) of the Bernoulli number to the ψ-component , and q · ω q (β(χ q )) ∈ Z × q . Proof. Part (a) is, stated with different notation, the same as Theorem 2 in [14, Ch. 1, §10]. The first assertion of part (b) is Remark 1 in [14, Ch. 1, §10] following the same theorem. For the second assertion of (b), note that by Proposition 4.3 we have F ker χq = Q(ζ q ), so f (χ q ) = q, and therefore Here ω q (η t ) is a (q − 1)-th root of unity in Z q , and, by definition of ω q and η t , it maps to (t mod q) in Z/qZ. Hence, q · ω q (β(χ q )) is an element of Z q that maps to (q − 1 mod q) and therefore belongs to Z × q .
The following result describes, for each χ ∈Ĝ − , the finite χ(T )-module Cl F up to Jordan-Hölder isomorphism in terms of Bernoulli numbers. We let p q be the prime ideal of χ q (T ) introduced in Proposition 4.3. Proof. In case (a), one sees from Proposition 4.5(a) that for every non-zero prime ideal m = m p,ψ of χ(T ) the element β(χ) ∈ Q(χ(G)) has non-negative valuation at m, so one has β(χ) ∈ χ(T ). In case (b), Proposition 4.5 shows that the same assertion has the single exception m = p q = m q,ωq , and that β(χ) has valuation −1 at p q ; so in that case one has p q β(χ) ⊂ χ(T ). We are now ready to prove Theorem 1.1. We first recall the statement.

Proof. By Theorem 4.7, we have [T
Since for all q ∈ U , the class group of Q(ζ q−1 ) is trivial, the ideals p q from Proposition 4.3 are principal for all q ∈ U , so the result follows from Proposition 4.3.
An analytic interlude. The aim of this subsection is to prove Theorem 4.10. This is a generalisation of [4, Corollary 3.7] from the trivial character to arbitrary Dirichlet characters. We will be brief, since the arguments are almost identical to those in [4] and [6].
Let M(T ) be a set of finite T -modules such that for every finite T -module M there is a unique M ′ ∈ M(T ) satisfying M ∼ = M ′ . For M ∈ M(T ), s = (s χ ) χ∈Ĝ/∼ ∈ CĜ /∼ , and u = (u χ ) χ∈Ĝ/∼ ∈ (Z ≥0 )Ĝ /∼ , we recall the following definitions from [4] and [6]: where Q is a projective T -module such that dim Q(χ(G)) (Q(χ(G))⊗ T Q) = u χ for all χ ∈Ĝ; it is easy to see that S u (M ) is well-defined, i.e. does not depend on the choice of Q; where we recall that the notation u → ∞ was defined at the beginning of Section 2. Observe that for u ∈ (Z ≥0 )Ĝ /∼ , one has |M | u = # Hom(Q, M ), where Q is as in the definition of S u . Moreover, if for all χ ∈Ĝ the ranks u χ are "large", then "most" homomorphisms Q → M , in a precisely quantifiable sense, are surjective. Making this precise, one deduces that w ∞ (M ) = 1/# Aut(M ). If u and s are as above, and f : M(T ) → C is any function, we define the following quantities when the respective limit exists: where 1 denotes the function M → 1 for all M ∈ M(T ), and M(T ) ≤B is defined analogously to M ≤B from Section 2. If T ′ is a quotient of T , we also define Z T ′ u (f, s) analogously to Z u (f, s), and Z T ′ (f, s) analogously to Z(f, s), but with the sums running only over T -modules that factor through T ′ , i.e. that are annihilated by the kernel of the quotient map T → T ′ , and we again set Z T ′ (s) = Z T ′ (1, s). With these definitions, the limit in (2.3) can be rewritten as where s P = (dim Q(χ(G)) (Q(χ(G)) ⊗ T P )) χ∈Ĝ/∼ , provided that the limit lim s→s P Z R (f, s) exists and is finite and that Z R (s P ) = 0. We stress that this is true even if the infinite sum Z R (s P ) diverges, in which case both the limit in (2.3) and that in (4.9) are equal to 0.
If T ′ = χ(T ) for some χ ∈Ĝ/∼, then Z T ′ u (f, s) as a function of s depends only on the entry s χ of s, and similarly for u, so we will write Z χ(T ) uχ (f, s χ ) in this case. In particular, if f is multiplicative in direct sums of the form M = χ∈Ĝ/∼ M χ , where for each χ ∈Ĝ/∼, the summand M χ is a χ(T )module, then one has Recall from the discussion at the beginning of the section that for each χ ∈Ĝ we have a canonical isomorphism G(χ(T )) ∼ = Cl χ(T ) ⊕ Z, and that if M is a finite χ(T )-module, then [M ] ∈ G(χ(T )) is contained in the torsion subgroup Cl χ(T ) of G(χ(T )).
where L Using the fact that f is multiplicative in short exact sequences of modules, one deduces from these two identities and a short calculation that there is a formal identity of Dirichlet series Z u+v (f, s) = Z u (f, s) · Z v (f, u + s). In particular, if 1 χ ∈ (Z ≥0 )Ĝ /∼ denotes the element that has χ-entry 1 and all other entries equal to 0, then A direct calculation shows that for each χ ∈Ĝ, one has Z χ , s + 1). It follows that for all u ∈ (Z ≥0 )Ĝ /∼ , one has a formal identity between Dirichlet series Q(χ(G)) (φ −1 χ , s+ 1) converges for Re s > τ χ , and for all k ∈ Z ≥2 , the Dirichlet series for L (S) Q(χ(G)) (φ −1 χ , s + k) converges absolutely for Re s > τ χ . It follows from this and from a classical result of Landau on convergence of products of Dirichlet series (see [10,Theorem 54]) that equation (4.11) is an equality of analytic functions, valid whenever Re s χ > τ χ for all χ ∈Ĝ ′ /∼. Finally, since, for every s ∈ C with Re s > τ χ , one has L (S) , we may take the limit of equation (4.11) as u → ∞, and the result follows.
Cohen-Martinet heuristics. We will now show that the conclusion of Corollary 4.8 contradicts the predictions of Heuristic 2.1.
Let G be a cyclic group of order 58, let S contain all prime numbers except 2 and 29, and let c ∈ G be the unique element of order 2. With these choices, if F/Q is a Galois extension with Galois group isomorphic to G such that 2) is equal to 1. This is not what Heuristic 2.1 predicts, as we will now explain.
One can check that the class group Cl R is equal to the class group Cl Z[ζ 29 ] , which is elementary abelian of order 8. By Theorem 4.10, the limit (2.3) with the above choices is equal to Q(ζ 29 ) (s) denotes the Dedekind zeta function of the field Q(ζ 29 ) with the Euler factors at primes above 2 and 29 omitted. When φ is non-trivial, the pole of the zeta function at s = 1 ensures that this limit is equal to 0 (see e.g. [11,Ch. 8]), and in particular is not equal to the limit (2.2).
One can deduce from the above calculation that the heuristics predict that the modules R ⊗ Z[G] Cl F should be equidistributed in Cl Z[ζ 29 ] as (F, ι) runs over the family F in Heuristic 2.1. Corollary 4.8 shows, however, that they almost all represent the trivial class.

Arakelov class groups of real abelian fields
In the present section we reinterpret Theorem 4.7 in terms of so-called oriented Arakelov class groups, proving Theorem 1.4 along the way. This reinterpretation leads to a more general result, Theorem 5.4, which pertains to all finite abelian extensions of Q, and which might conceivably be true in much greater generality; see Question 5.5.
We will use the generalities on group rings explained at the beginning of Section 4. In particular, G still denotes a finite abelian group, and we retain the notation S, T ,Ĝ.
Assume that S does not contain any prime numbers dividing 2 · #G. Let F be a real abelian number field with Galois group G over Q, and let O denote the ring of integers of , and ζ f denotes an arbitrarily chosen primitive f -th root of 1 in an algebraic closure of F . Let D be the subgroup of F × generated by the G-orbits of α k for all subfields k = Q of F that are cyclic over Q. We follow Mazur-Wiles [14] in defining the group of cyclotomic units of Proof. We will use without further explicit mention the facts that the ring Z (S) is flat over Z, and, for every χ ∈Ĝ, the ring χ(T ) is flat over T .
By Theorem 1 in [14, Ch. 1, §10], the subgroup C has finite index in O × . It follows that the annihilator of C (S) in T is generated by g∈G g ∈ T . The assertion of the proposition is therefore equivalent to the statement that for every non-trivial χ ∈Ĝ, the module χ(T )⊗ T C (S) is free of rank 1 over χ(T ), which we will now prove.
First, suppose that χ ∈Ĝ is faithful, non-trivial. In particular, G is nontrivial, cyclic. Then for all k F distinct from Q, we have χ(T )⊗ T T α k = 0, since all elements of T α k are fixed by a proper subgroup of G, while χ is faithful. Since the image of Norm F/Q ∈ T in the quotient χ(T ) is 0, we have Now, we deduce the general case. Let χ ∈Ĝ be arbitrary non-trivial, and let F ′ ⊂ F be the fixed field of ker χ. Temporarily write C F,(S) = C (S) , and let C F ′ ,(S) = Z (S) ⊗ Z C F ′ , where C F ′ is the analogously defined group of cyclotomic units of F ′ . The image of the element Norm F/F ′ = g∈ker χ g of T in the quotient χ(T ) is [F : F ′ ], which is invertible in χ(T ). It follows that χ(T ) ⊗ T C F,(S) = χ(T ) ⊗ T Norm F/F ′ (C F,(S) ). Moreover, it follows from a direct calculation, which we leave to the reader, that Norm F/F ′ C F,(S) ⊂ C F ′ ,(S) , and in fact, we have equality, since Norm F/F ′ acts as [F : F ′ ], and thus invertibly, on C F ′ ,(S) ⊂ C F,(S) . In summary, we have χ(T ) ⊗ T C F,(S) = χ(T ) ⊗ T C F ′ ,(S) . The assertion therefore follows from the special case proved above, applied to F ′ in place of F .
The map G → G given by g → g −1 for all g ∈ G extends Z (S) -linearly to an isomorphism between T and its opposite ring T opp . If M is any T -module, then the above isomorphism makes Hom(M, Z (S) ) and Hom(M, Q/Z) into T -modules, which we denote by M * and M ∨ , respectively.
We can now prove Theorem 1.4 from the introduction. Let us recall the statement.
The theorem therefore follows from Proposition 5.1. Ar F , and the theorem implies that the latter is 0. Theorems 5.2 and 4.7 can be elegantly combined into one statement, using the so-called oriented Arakelov class group of a number field F , defined by Schoof in [18]. To explain this, we will briefly recall the definition and the most salient properties of the oriented Arakelov class group, and refer the reader to [18] for details.
Let F/Q be a finite extension, let Id F be the group of fractional ideals of O F , and let F R denote the R-algebra R ⊗ Q F . The maximal compact where r and c denote the set of real, respectively complex places of F . It contains the group µ F of roots of unity of F as a subgroup. We have canonical maps Id F → R >0 and F × R → R >0 , the first given by the ideal norm, and the second given by the absolute value of the R-algebra norm. Let Id F × R >0 F × R be the fibre product with respect to these maps. The oriented Arakelov class group Pic 0 F of F is defined as the cokernel of the map F × → Id F × R >0 F × R that sends α ∈ F × to (αO F , α). It is a compact abelian Aut F -module, whose dual Hom( Pic 0 F , R/Z) will be denoted by Ar F . One has an exact sequence of finitely generated discrete Aut F -modules 0 → Ar F → Ar F → Hom((F × R )/µ F , R/Z) → 0. Let K be an algebraic number field, let F/K be a finite Galois extension, let G be its Galois group, let S is any set of odd prime numbers, let Σ be the set of infinite places of K, and for v ∈ Σ let I v ⊂ G denote an inertia subgroup at v. Then one has an isomorphism of Z (S) [G]-modules where Ind G/Iv denotes the induction from I v to G, and Let d be the degree of K over Q. By combining the above observation with the exact sequence (1.3), we deduce the equalities and hence denotes the permutation module with a Z (S) -basis given by the set of cosets G/I v , and with G acting by left multiplication. Note that this equality holds for all finite Galois extensions, not just abelian ones. From this equality, together with Theorems 5.2 and 4.7, one easily deduces the following result. The theorem suggests the following question.
Question 5.5. Let K be an algebraic number field, let d be its degree over Q, let F/K be a finite Galois extension, let G be its Galois group, let S be a set of prime numbers not dividing 2 · #G, and let T = Z (S) [G]. Is the class of Equation (5.3) shows that the answer to Question 5.5 is affirmative when the natural map G(T ) → G(Q ⊗ Z (S) T ) is injective, which is the case for example when S is finite, as can be deduced from Lemma 3.5 and [16, Theorem (21.4)].

Enumerating number fields
In this section, we give our second disproof of the Cohen-Martinet heuristics. We begin by proving Theorem 1.2 from the introduction, and then compare its statement with the predictions of Heuristic 2.1. Our disproof suggests that the discriminant is not a good invariant to use for purposes of arithmetic statistics, and we investigate alternatives.
LetQ be a fixed algebraic closure of Q, and let C(x) be the set of all fields F ⊂Q that are Galois over Q with cyclic Galois group of order 4 and whose discriminant is at most x. If k ⊂Q is a quadratic field, let C k (x) = {F ∈ C(x) : k ⊂ F }. If n is a positive integer, let σ 0 (n) denote the number of positive divisors of n.
where p ranges over primes. Let k ⊂Q be a quadratic field, and let d be its discriminant. If d < 0 or d has at least one prime divisor that is congruent to 3 (mod 4), then for all real numbers x, the set C k (x) is empty. In all other cases, the limit exists, and is equal to if d is odd, respectively even, where in both expressions p ranges over primes.
Proof. We will use the estimates of [15]. If F is a cyclic quartic field containing k, then any place of Q that ramifies in k must be totally ramified in F , so k must be real, and only primes that are congruent to 1 or 2 (mod 4) can ramify in k. This proves the first assertion.
In the rest of the proof, assume that d is positive and not divisible by any primes that are congruent to 3 (mod 4). Write d = 2 β d ′ , where β ∈ {0, 3}, and d ′ is a product of distinct prime numbers that are congruent to 1 (mod 4). Then the discriminant of any cyclic quartic field containing k is of the form n = 2 α d ′3 a 2 , where a is an odd square-free positive integer that is coprime to d ′ , and α = 11 if β = 3, and α ∈ {0, 4, 6} if β = 0 (see [15]). Note that k = Q( √ n), so the discriminant of a cyclic quartic field determines its unique quadratic subfield. Let h(n) be the number of cyclic quartic fields insideQ of discriminant n. By [15, equation (3.3)] we have For a real number x, let γ(x) denote the number of square-free positive integers that are at most x and coprime to 2d. It follows from the above discussion, that if d is odd, then where the sums run over square-free positive integers a that are coprime to 2d; while if d is even, then with the sum again running over square-free positive integers a that are coprime to 2d. A standard estimate shows that for a positive integer m, we have where p ranges over primes. Combining this estimate with the above formulae for #C k (x) yields if d is even, where again p ranges over primes. On the other hand, by [15] we have , whence the result follows.
We can now prove Theorem 1.2 from the introduction. We recall the statement. Proof. For a quadratic field k and a positive real number x, let p k (x) = #C k (x)/#C(x), so that lim x→∞ p k (x) is as described by Theorem 6.1.
Let K be any set of real quadratic fields. Then we claim that lim x→∞ k∈K where the last equality follows from the fact that all the summands are non-negative. An easy calculation with the values for lim x→∞ p k (x) given by Theorem 6.1 shows that k lim x→∞ p k (x) = 1, where the sum runs over all real quadratic fields. By combining this with inequality (6.4) applied to the complement of K in the set of all real quadratic fields in place of K, one deduces that lim sup x→∞ k∈K p k (x) ≤ k∈K lim x→∞ p k (x), whence the claimed equality (6.3) follows.
When applied to the set K of real quadratic fields whose class number is not divisible by 3, this shows that the limit lim x→∞ #C ′ (x) #C(x) exists. Moreover, by summing lim x→∞ p k (x), as given by Theorem 6.1, over those real quadratic fields of discriminant less than 3.1 × 10 9 whose class number is, respectively is not, divisible by 3 one obtains sufficiently tight upper and lower bounds on that limit to obtain the estimate in the theorem. then the value of (2.2) is nothing but the limit lim x→∞ #C ′ (x)/C(x) referred to in Theorem 6.2. The same argument as in the proof of the theorem shows that, more generally, if one chooses A and G as just described, S to be any set of odd primes, the projective module P to have rank 1 over R, and f to be any computable bounded C-valued function on M, then the limit (2.2) exists and can be computed to any desired precision. Theorem 6.2 contradicts the predictions of Heuristic 2.1, as we will now explain. As just remarked, the limit lim x→∞ #C ′ (x)/C(x) of Theorem 6.2 is equal to the value of (2.2) for suitable choices of G, A, S, P , and f . The value of (2.3), on the other hand, with these choices is the same as with a different choice, namely as with G ′ = h|h 2 = id , A ′ = Q[G ′ ]/(1 + h) ∼ = A, and S ′ = S, P ′ = P , and f ′ = f . With that latter choice, the value of (2.3) was computed in [5, (1.2)(b)] to be ≈ 0.8402, and thus not equal to the value of (2.2), which is given by Theorem 6.2. This completes our disproof of Heuristic 2.1.
The above disproof relies on the observation that enumerating number fields by non-decreasing discriminant has the undesirable feature that certain fields may appear with positive probability as intermediate fields, so that Heuristic 2.1 for those number fields clashes with that for the intermediate fields. We conjecture that instead enumerating fields by the ideal norm c F/K of the product of the primes of O K that ramify in F/K does not exhibit this feature. The following is a result in support of this conjecture. Recall that if F is a set of pairs (F, ι) as in the introduction, where all fields F are Galois extensions of a given field K, then for every positive real number B we define F c≤B = {(F, ι) ∈ F : c F/K ≤ B}. Proposition 6.6. Let K be a number field, letK be an algebraic closure of K, let G be a finite abelian group, let V be a finitely generated Q[G]-module, and let F be the set of all pairs (F, ι), where F ⊂K is a Galois extension of K, and ι is an isomorphism between the Galois group of F/K and G that induces is not empty. Let k ⊂K be a field properly containing K. Then the limit lim B→∞ #{(F, ι) ∈ F c≤B : k ⊂ F }/#F c≤B is zero.
Proof. Let T be an infinite set of prime ideals of O K with odd residue characteristics that are not totally split in k/K. Then for any F ⊂K that contains k, and for all p ∈ T , we have F ⊗ K K p ∼ = K #G p as K p -algebras, where K p denotes the field of fractions of the completion of O K at p. It follows that for all positive real numbers B, we have #{(F, ι) ∈ F c≤B : k ⊂ F } ≤ #{(F, ι) ∈ F c≤B : ∀p ∈ T : F ⊗ K K p ∼ = K #G p }. By [13, Theorem 2.1], for every finite subset T ′ of T , each of the following limits exists, and there is an equality moreover, [13, Theorem 2.1] also implies that each of the factors on the right hand side is bounded away from 1 uniformly for all p ∈ T . Thus, we have lim sup as claimed.

Reasonable functions
In the present section we address the question of which functions may qualify as "reasonable" for the purposes of Conjecture 1.6. We begin by proving Theorem 1.7, which suggests that demanding that E(|f |) should exist is likely not a sufficient criterion. After that, we offer two possible interpretations of the word "reasonable".
From now on, let (X, p) be an infinite discrete probability space such that p(x) > 0 for all x ∈ X, where we recall from the introduction that p(x) is shorthand for p({x}), and let Y be the probability space X Z ≥1 with the induced product probability measure; see e.g. [9, §38]. When referring to subsets of a measure space, we will say "almost all" to mean a subset whose complement has measure 0.
Let us reformulate Theorem 1.7, using the above notation.
Theorem 7.1. For almost all sequences y = (y i ) i ∈ Y , there exists a function f : X → R ≥0 whose expected value E(f ) is finite, but for which the average lim n→∞ 1 n n i=1 f (y i ) of f on y does not exist in R. The idea of the proof will be to show that, for a typical (y i ) i ∈ Y , there are many elements x ∈ X that occur much earlier in (y i ) i than one would expect. The function f then gives those elements a large weight.
Proposition 7.2. For almost all sequences y = (y i ) i ∈ Y it is true that for all ǫ ∈ R >0 there exist infinitely many x ∈ X such that for some i ≤ ǫ/p(x), one has y i = x.
Proof. Let ǫ ∈ R >0 be given, and let U be a finite subset of X. First, we claim that for almost all sequences y = (y i ) i ∈ Y there exists x ∈ X \ U such that for some i ≤ ǫ/p(x), one has y i = x. If x is an element of X, let E x be the set of y = (y i ) i ∈ Y for which y i = x for all i ≤ ǫ/p(x). The probability of the event E x , meaning the measure of E x ⊂ Y , is equal to (1 − p(x)) ⌊ǫ/p(x)⌋ , which tends to e −ǫ , as p(x) tends to 0, and in particular is uniformly bounded away from 1 for all but finitely many x ∈ X. If x 1 , x 2 , . . . , x k are distinct elements of X, then the events E x i are generally not independent, but the probability of E x k given that all of E x 1 , . . . , E x k−1 occur is clearly less than or equal to the probability of E x k . It follows that the probability that E x occurs for all x ∈ X \ U is at most x∈X\U (1 − p(x)) ⌊ǫ/p(x)⌋ = 0. This proves the claim.
Since the number of finite subsets U of X is countable, it follows from countable subadditivity that if ǫ ∈ R >0 is given, then for almost all y = (y i ) i ∈ Y it is true that for all finite subsets U of X there exists x ∈ X \ U such that for some i ≤ ǫ/p(x), one has y i = x. This implies that if ǫ ∈ R >0 is given, then for almost all y = (y i ) i ∈ Y there exist infinitely many x ∈ X such that for some i ≤ ǫ/p(x), one has y i = x. By applying this conclusion to countably infinitely many ǫ n ∈ R >0 in place of ǫ, where (ǫ n ) n∈Z ≥1 is a sequence converging to 0, and by invoking countable subadditivity again, we deduce the proposition.
We are now ready to prove Theorem 7.1.
Proof of Theorem 7.1. Let y = (y i ) ∈ Y be a sequence for which the conclusion of Proposition 7.2 holds. Then there is a sequence x 1 , x 2 , . . . of distinct elements of X such that for each n ∈ Z ≥1 , we have min{i ∈ Z ≥1 : y i = x n } ≤ n −3 /p(x n ).
For n ∈ Z ≥1 , let i(n) = min{j ∈ Z ≥1 : y j = x n } ≤ n −3 /p(x n ). For x ∈ X, define f (x) = 0 if x = x n for any n ∈ Z ≥1 , and f (x n ) = n −2 /p(x n ). Then we have E(f ) = n∈Z ≥1 n −2 , which converges. On the other hand, for every n ∈ Z ≥1 , one has which gets arbitrarily large, as n varies, so the limit lim n→∞ 1 n n j=1 f (y j ) does not exist.
Let us now discuss two possible interpretations of the word "reasonable" in Conjecture 1.6. As will hopefully become clear, this section should be treated as an invitation to the reader to join in our speculations. If M V is as in Conjecture 1.6, let f : M V → C be a function. For a positive integer i, the i-th moment of f is defined to be M ∈M V (f (M ) i P(M )) if the sum converges absolutely.
Conjecture (Supplement 1 to Conjecture 1.6). If f : M V → C is a function such that for all i ∈ Z ≥1 , the i-th moment of |f | exists in R, then f is "reasonable" for the purposes of Conjecture 1.6.
All bounded functions on M V , and many unbounded functions of arithmetic interest satisfy this condition. This applies to all examples in [5], with the exception of the functions f (M ) = #M , and f (M ) = (#M ) 2 , which often have expected value ∞. Note that in [5] the set S of prime numbers was not assumed to be finite. Here, when we talk about the examples in [5], we mean the analogues in our setting of the functions considered there. On the other hand, it can be shown that if X = M V , A = 0, and S is non-empty, so that M V is infinite, then for the function constructed in the proof of Theorem 7.1 the second moment does not exist.
Let us call a class R of C-valued functions on X promising if for all f ∈ R, the expected value E(f ) exists, and for almost all sequences y = (y j ) ∈ Y it is true that for all f ∈ R we have lim n→∞ 1 n n j=1 f (y j ) = E(f ). By the law of large numbers, for any function f : X → C for which E(f ) exists, the set R = {f } is promising. It immediately follows that any countable set R of such functions is promising. On the other hand, Theorem 7.1 implies that the class of all functions f : X → C for which E(f ) exists is not promising. An affirmative answer to the following question would strengthen our confidence in Supplement 1.  Proof. For any c ∈ R >0 , let R c be the class of all functions f : X → C for which sup x∈X |f (x)| ≤ c. We will first show that for all c ∈ R >0 , the class R c is promising.
Let X 1 ⊂ X 2 ⊂ . . . be a sequence of finite subsets of X such that lim i→∞ x∈X i p(x) = 1. Fix c ∈ R >0 , and let ǫ ∈ R >0 be arbitrary. Then we may choose X i(ǫ) such that x∈X\X i(ǫ) p(x) ≤ ǫ/c. Moreover, by the strong law of large numbers, there exists a subset Y c (ǫ) ⊂ Y of measure 1 with the following property: for all y = (y j ) ∈ Y c (ǫ) there is an N y (ǫ) ∈ Z ≥1 such that for all x ∈ X i(ǫ) and for all n ≥ N y (ǫ) one has | 1 n · #{j ≤ n : y j = x} − p(x)| ≤ ǫ/(c · #X i(ǫ) ). It follows that for all y ∈ Y c (ǫ), for all f ∈ R c , and for all n ≥ N y (ǫ) one has 1 n n j=1 f (y j ) − E(f ) = x∈X 1 n · #{j ≤ n : y j = x} − p(x) · f (x) ≤ x∈X i(ǫ) 1 n · #{j ≤ n : y j = x} − p(x) · f (x) Therefore, if (ǫ n ) n∈Z ≥1 is a sequence of positive real numbers converging to 0, then the intersection Y c = n∈Z ≥1 Y c (ǫ n ) has measure 1 and has the property that for every y = (y j ) ∈ Y c and for every f ∈ R c one has lim n→∞ 1 n n j=1 f (y j ) = E(f ). If (c n ) n∈Z ≥1 is a sequence of positive real numbers tending to ∞, then the class of all bounded functions is equal to n∈Z ≥1 R cn , and the intersection n∈Z ≥1 Y cn has measure 1 and has the property that for every sequence y = (y i ) in this intersection and for every bounded function f one has lim n→∞ 1 n n i=1 f (y i ) = E(f ). This completes the proof.
We will now describe a completely different approach to the question of reasonableness, which is based on the idea that one can distinguish between "highly artificial" functions, such as those that are constructed in the proof of Theorem 7.1, and "natural" functions that one cares about in practice by the ease with which they can be computed.
Let A, S, R, and V be as in Conjecture 1.6, and suppose that S is nonempty and A = 0. Let Q be the set of non-increasing sequences (n i ∈ Z ≥0 : i ∈ Z ≥0 ) that have only finitely many non-zero terms. Let Maxspec(Z(R)) be the finite set of maximal ideals of the centre Z(R) of R. It follows from [6, Lemme 2.7] that the set M is canonically in bijection with the set Q Maxspec(Z(R)) , and by Lemma 3.5 we also have a bijection between M V and Q Maxspec(Z(R)) .
Conjecture (Supplement 2 to Conjecture 1.6). If f : M V → Z is a function such that E(|f |) exists in R, and the induced function Q Maxspec(Z(R)) → Z is computable in polynomial time, where the input is given in unary notation, then f is "reasonable" for the purposes of Conjecture 1.6.
The functions that one typically cares about in practice, including all those given as examples in [5] for which S is finite, are computable in polynomial time. On the other hand, we do not expect the function that was constructed in the proof of Theorem 7.1 for y being the sequence of class groups in a family of number fields to be computable in polynomial time. Indeed, to define f (M ), one needs to know roughly the first # Aut(M ) terms of the sequence y, and # Aut(M ) is exponential in the size of the input.