Wild Partitions and Number Theory

David P. Roberts Division of Science and Mathematics University of Minnesota, Morris Morris, MN, 56267 USA

                    roberts@morris.umn.edu

Abstract

We introduce the notion of wild partition to describe in combinatorial language an important situation in the theory of p-adic fields. For Q a power of p, we get a sequence of numbers λ_{Q,n} counting the number of certain wild partitions of n. We give an explicit formula for the corresponding generating function Λ_{Q}(x) = ∑ λ_{Q,n}x^{n} and use it to show that λ_{Q,n}^{1/n} tends to Q^{1/(p−1)}. We apply this asymptotic result to support a finiteness conjecture about number fields. Our finiteness conjecture contrasts sharply with known results for function fields, and our arguments explain this contrast.

1 Introduction

The sequence \underline{A000041} of integers λ_{n} giving the number of partitions of n is important throughout mathematics. Its generating function is

Λ(x) = ∑_{n=0}^{∞} λ_{n}x^{n} = ∏_{e=1}^{∞} \frac{1}{1 − x^{e}} = 1 + x + 2x^{2} + 3x^{3} + 5x^{4} + 7x^{5} + · · · .  (1)

In this paper, we consider for any prime power Q = p^{n_{0}}, an analogous integer sequence λ_{Q,n} arising in a fundamental way in the number theory of p-adic fields. We evaluate the associated generating functions Λ_{Q}(x) = ∑ λ_{Q,n}x^{n}, obtain corresponding asymptotics, and apply our results to support a finiteness conjecture about number fields. Generally speaking, our goal is to describe how a combinatorial viewpoint clarifies an important number-theoretic situation.

The following two displays, incorporated in the sequence database [24] as \underline{A131139} and \underline{A131140}, give a first sense of the functions Λ_{Q}(x) and the corresponding sequences λ_{Q,n}:

Λ_{2}(x) = \frac{1}{1 − x} · \frac{1 − x^{2}}{(1 − 2x^{2})^{2}} · \frac{1}{1 − x^{3}} · \begin{matrix}(1 − x^{4}) & (1 − 4x^{4}) & 2 \\ (1 − 8x^{4}) & 4 \end{matrix} · \frac{1}{1 − x^{5}} · \frac{1 − x^{6}}{(1 − 8x^{6})^{2}} ⋯
= 1 + x + 4x^{2} + 5x^{3} + 36x^{4} + 40x^{5} + 145x^{6} + 180x^{7} + 1572x^{8} + 1712x^{9} + ⋯,

Λ_{3}(x) = \frac{1}{1 − x} · \frac{1}{1 − x^{2}} · \begin{matrix}(1 − x^{3}) & 2 \\ (1 − 3x^{3}) & 3 \end{matrix} · \frac{1}{1 − x^{4}} · \frac{1}{1 − x^{5}} · \begin{matrix}(1 − x^{6}) & 2 \\ (1 − 9x^{6}) & 3 \end{matrix} ⋯
= 1 + x + 2x^{2} + 9x^{3} + 11x^{4} + 19x^{5} + 83x^{6} + 99x^{7} + 172x^{8} + 1100x^{9} + · · · .

In general, Λ_{Q}(x), like its model Λ(x), is given by a product over positive integers e. For p not dividing e, the corresponding factor is 1/(1 − x^{e}) again. However for p dividing e, this factor is more complicated. In the number-theoretic context, the former factors reflect tame ramification and the latter reflect wild ramification.

The paper is organized so its starts in combinatorics and ends in number theory. The main combinatorial objects, wild partitions, are defined so that they correspond bijectively to the main number theoretic objects, geometric classes of p-adic algebras. We do not pursue it here, but a future goal is to specify one bijection as the conventional one, so that the very simple objects, wild partitions, index the more complicated objects, geometric classes of p-adic algebras. Such a labeling of p-adic algebras could be incorporated into the database of local fields [12] and would considerably facilitate the p-adic analysis of number fields.

Sections 2-4 are our combinatoric sections. For a factorization n_{0} = e_{0}f_{0}, we define (p, e_{0}, f_{0})-wild partitions and a corresponding complicated three-variable generating function Λ_{p,e_{0},f_{0}}(x, y, z). Our definitions are not particularly motivated from a purely combinatoric point of view. Rather, as indicated above, they are chosen to mimic the structure of p-adic fields. For the sake of comparison, we consider first the specialization (y, z) = (1, p^{−f_{0}}) and get the remarkable simplification

Λ_{p,e_{0},f_{0}}(x, 1, p^{−f_{0}}) = Λ(x).  (2)

As we’ll indicate, this identity is related to the Serre mass formula [23]. Our main interest is in the new specialization (y, z) = (1, 1). We define Λ_{Q}(x) by an explicit formula and find

Λ_{p,e_{0},f_{0}}(x, 1, 1) = Λ_{Q}(x),  (3)

independently of the factorization n_{0} = e_{0}f_{0}. Our explicit formula allows us to consider arbitrary real powers Q = p^{ν} ≥ 1 so that Q no longer determines p and we accordingly write Λ_{p,Q}(x). One has Λ_{p,1}(x) = Λ(x), independently of p. Thus another point of view is that for each prime p we have a Q-analog of Λ(x).

Sections 5 and 6 consider analytic number theory associated to Λ_{p,Q}(x). We express Λ_{p,Q}(x) directly in terms of Λ(x) and observe that as a consequence

lim_{n→∞} λ_{p,Q,n}^{1/n} = Q^{1/(p−1)}.  (4)

Thus λ_{p,Q,n} grows exponentially with growth factor Q^{1/(p−1)}. Equation (4) contrasts with the famous Hardy-Ramanujan statement [8] of subexponential growth, λ_{n} ∼ e^{π\sqrt{2n/3}}/(4n\sqrt{3}). It quantifies the extent to which wild ramification predominates over tame ramification in number theory. Another contrast between ordinary partitions and our Q-analogs is that λ_{n}/λ_{n−1} tends to 1 while λ_{p,Q,n}/λ_{p,Q,n−1} has oscillatory behavior which becomes more pronounced as Q increases. We conjecture an asymptotic of the form

λ_{p,Q,n} ∼ c_{p,Q}(n)C_{p}(Q)n^{B_{p}(Q)}e^{A_{p}(Q)\sqrt{n}}Q^{n/(p−1)},  (5)

with an explicit factor c_{p,Q}(n) capturing the oscillatory behavior of λ_{p,Q,n}/λ_{p,Q,n−1}.

Sections 7-10 are set in the framework of local algebraic number theory. The material here is somewhat more technical, but we have arranged our presentation so that the only prerequisite is familiarity with basic facts about p-adic fields. Section 7 sets up the general situation and illustrates it for the fields ℝ and ℂ, getting simple functions Λ_{ℝ}(x) = e^{x} and Λ_{ℂ}(x) = e^{x+x^{2}/2} which serve as analogs of our Λ_{Q}(x). For Sections 8-10, we let F be an extension field of the p-adic field ℚ_{p}, of ramification index e_{0}, inertial degree f_{0}, and thus degree n_{0} = e_{0}f_{0} and residual cardinality q = p^{f_{0}}. Section 9 explains how the coefficient λ_{n,c_{t},c_{w}} of x^{n}y^{c_{t}}z^{c_{w}} gives the “total mass” of algebras K over F having relative degree n, tame conductor c_{t}, and wild conductor c_{w}. Section 10 works over the maximal unramified extension F^{un} of F. It explains how λ_{n,c_{t},c_{w}} also counts extension algebras of F^{un} with the corresponding invariants. The perspective of Section 9 is more directly connected with the literature, while the perspective of Section 10 explains why the coefficients of Λ_{p,e_{0},f_{0}}(x, y, z) are integers. Summing over the possible c_{t} and c_{w}, one gets that the total mass λ_{F,n} of degree n extension algebras of F is λ_{Q,n}, where Q = p^{n_{0}}.

Section 11 shifts to global algebraic number theory, working over an arbitrary number field F. It addresses a question raised in [16] on the size of sets Fields_{F,n,S}^{big} of relative degree n number fields K/F. To be in Fields_{F,n,S}^{big}, the extension K/F must have associated Galois group A_{n} or S_{n} and ramification contained within the prescribed finite set of places S of F including the Archimedean places. A recent heuristic of Bhargava [4] yields

\frac{1}{2} ∏_{v∈S} λ_{F_{v},n}  (6)

as a first guess (after slight modifications in degrees ≤ 3) for the size of Fields_{F,n,S}^{big}. The Archimedean factors λ_{ℝ,n}, λ_{ℂ,n} decay superexponentially and we have proved that the remaining λ_{F_{v},n} grow only exponentially. Thus (6) leads to the prediction that for fixed (F, S), the set Fields_{F,n,S}^{big} is empty for sufficiently large n. In other words Fields_{F,S}^{big} = ∐_{n} Fields_{F,n,S}^{big} is finite. This finiteness statement has a certain irony to it: normally one considers A_{n} and especially S_{n} to be the “generic expectation” for Galois groups of number fields; the statement says that in the setting of prescribed ramification, these groups are in fact the exceptions.

Finally, Section 12 considers positive characteristic analogs of all the previous considerations. In positive characteristic, our finiteness statement fails very badly. The theory we present here explains this failure as due to two sources, either one of which suffices to void our argument for the finiteness statement. One source is that Q must be considered ∞ and this forces all the λ_{F_{v},n} appearing in (6) to be infinite for n ≥ p. Another source is that there are no Archimedean places of F, and thus no superexponentially decaying factors in (6).

Readers who want to quickly see the main ideas in a streamlined setting are invited to first focus on the special case e_{0} = f_{0} = 1. Then n_{0} = 1 too, p = q = Q, and later E = e. The ground fields F of Sections 7-10 are then only the completions of ℚ, i.e. ℝ and the ℚ_{p}. The ground fields F of Section 11 are then limited to simply ℚ itself. However by explicit examples with n_{0} = 2 in Sections 2, 3, 6, and 11, we try to assist readers in appreciating the case of general (e_{0}, f_{0}). Sections 7-11 make clear that general (e_{0}, f_{0}) is the natural setting from a number-theoretic point of view. The naturality of this setting is emphasized by Sections 5 and 6 which interpolate n_{0} = e_{0}f_{0} ∈ ℤ_{≥1} with general reals ν ≥ 0. The naturality of the general setting is further underscored by Section 12, which is based on the limiting case e_{0} = ∞.

2 Wild partitions

Basic notation. An ordinary partition is an element of the free abelian monoid generated by the set of allowed parts P = {1, 2, 3, … }, for example

μ_{ordinary} = 9 + 7 + 3 + 3 + 2 + 2 + 2 + 1.  (7)

Wild partitions are more complicated in two ways. First, the set P is replaced by a set P (p, e_{0}) mapping surjectively to P, with infinite fibers above multiples of p. Second, necessary for obtaining finiteness, an invariance condition with respect to an operator σ = σ_{p}^{f_{0}} enters.

As just indicated, our notion of wild partition depends not only on a prime p, but also on two positive integers e_{0} and f_{0}. Let q = p^{f_{0}} and n_{0} = e_{0}f_{0}. Our notations e_{0}, f_{0}, n_{0} and q come from standard notations in the number-theoretic situation of Sections 8-10 inspiring our definitions. The quantity Q = p^{n_{0}} is important for us, but does not have a standard number-theoretic notation.

Strictly speaking, our sets of wild partitions depend also on a choice of algebraic closure \overline{𝔽}_{p} of the prime field 𝔽_{p} = ℤ/pℤ. However all algebraic closures are isomorphic and so our final formulas counting certain wild partitions are independent of this choice. As usual, for a power p^{u} of p we denote by 𝔽_{p^{u}} the unique subfield of \overline{𝔽}_{p} of cardinality p^{u}. We denote by σ_{p} the Frobenius element k ↦ k^{p} in Gal(\overline{𝔽}_{p}/𝔽_{p}). Similarly, we denote by σ_{p^{u}} the element σ_{p}^{u}; it is a topological generator of Gal(\overline{𝔽}_{p}/𝔽_{p^{u}}). Most important for us is the operator σ_{q}, which we abbreviate by simply σ.

We reserve e for our main variable running over P. As a standing convention, we systematically write e = p^{w}t with p^{w} the largest power of p dividing e. We think of w as the wildness of e, p^{w} as the wild part of e, and t as the tame part of e. As another standing convention, we abbreviate e_{0}e by E.

Ore numbers and their associated dimensions and spaces. An important notion in number theory is the set of Ore numbers

Ore(p, e_{0}, e) ⊆ {0, 1, …, wE − 1, wE}.  (8)

To understand the set Ore(p, e_{0}, e), it is convenient present it as an array, as for the case (p, e_{0}, e) = (3, 2, 9) for which w = 2:

\begin{matrix}. \\ . & 8 & 7 & . & 5 & 4 & . & 2 & 1 \\ . & 17 & 16 & . & 14 & 13 & . & 11 & 10 \\ . & 26 & 25 & 24 & 23 & 22 & 21 & 20 & 19 \\ 36 & 35 & 34 & 33 & 32 & 31 & 30 & 29 & 28 \end{matrix}  (9)

In general, the array Ore(p, e_{0}, e) consists of a degenerate zeroth block, followed by w full blocks. The zeroth block has only a single spot, filled by 0 if w = 0 and empty otherwise. The full blocks each have e_{0} rows and e columns. For 1 ≤ j ≤ w − 1, the j^{th} block consists of the integers in the interval [(j − 1)E + 1, jE] which are not multiples of p^{j}. The w^{th} block consists of these entries together with wE. Considering the table as a whole, we refer to all the entries as non-maximal, except for wE which is maximal. Our array format, including the right-to-left order, is intended to facilitate the discussion in Section 8, where the number-theoretic origin of Ore(p, e_{0}, e) is explained.

An important quantity in our situation is the dimension d(p, e_{0}, e, s) associated to an Ore number s ∈ Ore(p, e_{0}, e). It is the number of integers in [0, s − 1] which are not in Ore(p, e_{0}, e). Thus d(3, 2, 9, 20) = 7, as there are seven omitted numbers less than 20 on the displayed Ore table (9). The way dimensions arise in number theory is explained in Section 9.

An Ore number s ∈ Ore(p, e_{0}, e) determines a subset W (p, e_{0}, e, s) of the vector space \overline{𝔽}_{p}^{d(p,e_{0},e,s)} as follows. For non-maximal s, the set W (p, e_{0}, e, s) consists of the subset of vectors with non-zero first coordinate. In the maximal case s = wE, the subset W (p, e_{0}, e, s) is defined to be all of \overline{𝔽}_{p}^{d(p,e_{0},e,s)}. The Frobenius element σ = σ_{p}^{f_{0}} acts coordinate-wise on each W (p, e_{0}, e, s), as indeed σ_{p} itself acts. The number of fixed points of σ is clearly q^{d(p,e_{0},e,s)}(1 − 1/q) for non-maximal s and q^{d(p,e_{0},e,s)} for the maximal s = wE. The explicit σ-sets W (p, e_{0}, e, s) just introduced are isomorphic to less explicit σ-sets arising naturally in number theory, as explained in Section 10. We use s as our variable running over Ore numbers, because Ore numbers are also called Swan conductors.

Wild partitions and associated invariants. We are now in a position to make the main definition of the combinatorial part of this paper.

Definition 2.1. A (p, e_{0}, f_{0})-wild partition is an element of the free abelian monoid on the set

P (p, e_{0}) = ∐_{e∈ℤ_{≥0}} ∐_{s∈Ore(p,e_{0},e)} W (p, e_{0}, e, s)  (10)

which is fixed by σ = σ_{p}^{f_{0}}.

Usually (p, e_{0}, f_{0}) is fixed and clear from the context. Then we just say “wild partition” rather than (p, e_{0}, f_{0})-wild partition.

We denote elements of P (p, e_{0}) as doubly-subscripted integers e_{s;ω}, with s ∈ Ore(p, e_{0}, e) and ω ∈ W (p, e_{0}, e, s). If p does not divide e, then the only possible subscript is “0;0” and so we allow ourselves to omit it. As an example of our notation, let i be one of the two square roots of −1 in 𝔽_{9}. Then

μ_{wild} = 9_{20;1,1,0,2,2,0,1} + 7 + 3_{1;i} + 3_{1;−i} + 2 + 2 + 2 + 1  (11)

is a wild partition for (p, e_{0}, f_{0}) = (3, 2, 1). To check that μ_{wild} is indeed formed according to our rules, note that d(3, 2, 9, 20) = |{0, 3, 6, 9, 12, 15, 18}| = 7 from (9), and so it is proper that first subscripted ω has length 7. Also the first coordinate of this ω is non-zero, as required. The Ore table for (p, e_{0}, e) = (3, 2, 3) omits 0 and has first row “· 2 1”, so that d(3, 2, 3, 1) = |{0}| = 1; thus 3_{1;i} and 3_{1;−i} are properly constructed wild parts. Finally q = p^{f_{0}} = 3^{1} = 3 and so σ(i) = i^{3} = −i; thus μ_{wild} satisfies the σ-invariance condition.

By definition, one can add wild partitions just as one can add ordinary partitions. Wild partitions have three obvious additive integer invariants, all important in the underlying number-theoretic situation. First, as for ordinary partitions, one has the degree n, defined as usual as the sum of the parts e_{i}. Second, defined but often not important for ordinary partitions, one has the tame conductor c_{t}, the sum of the e_{i} − 1. Third, particular to our wild situation, one has the wild conductor c_{w}, the sum of the first subscripts s_{i}. Thus for the wild partition (11), one has (n, c_{t}, c_{w}) = (29, 21, 22).

3 The generating functions Φ_{F}(x, y, z) and Λ_{F}(x, y, z)

Fix for this section a triple (p, e_{0}, f_{0}) as in the previous section. These three quantities figure rather passively into our current considerations. We will have other more active quantities as well. Accordingly, we abbreviate via F = (p, e_{0}, f_{0}). When we are continuing the example started in (7), (11), we will take F = (3, 2, 1).

Irreducible and isotypical partitions. We say a wild partition is irreducible if it is nonzero and cannot be written as the sum of two non-zero wild partitions. Every wild partition is uniquely the sum of its irreducible constituents. For example, μ_{wild} has seven irreducible constituents,

μ_{wild} = 9_{20;1,1,0,2,2,0,1} + 7 + (3_{1;i} + 3_{1;−i}) + 2 + 2 + 2 + 1.  (12)

We similarly say that a wild partition is isotypical if it has the form mμ for μ an irreducible wild partition and m a positive integer. Every wild partition is uniquely the sum of its isotypical constituents. For example, μ_{wild} has five isotypical constituents,

μ_{wild} = 9_{20;1,1,0,2,2,0,1} + 7 + (3_{1;i} + 3_{1;−i}) + (2 + 2 + 2) + 1.  (13)

If the wild partition μ is irreducible, we say that the isotypical wild partition mμ has mass 1/m. In the number-theoretic settings of Sections 8-10, wild partitions correspond to geometric packets of algebras of total mass one. A packet contains fields if and only if the corresponding wild partition is isotypical; in this case, the fields in the packets have total mass 1/m and the non-fields total mass 1 − 1/m.

Definition of Φ_{F}(x, y, z) and Λ_{F}(x, y, z). Let ϕ_{F,n,c_{t},c_{w}} be the total mass of F-wild isotypical partitions of degree n, tame conductor c_{t}, and wild conductor c_{w}. Then the corresponding generating function is expressible as a sum over irreducibles,

Φ_{F}(x, y, z) = ∑_{n=0}^{∞} ∑_{c_{t}=0}^{∞} ∑_{c_{w}=0}^{∞} ϕ_{F,n,c_{t},c_{w}}x^{n}y^{c_{t}}z^{c_{w}}  (14)

= ∑_{μ} ∑_{m=1}^{∞} \frac{1}{m}x^{mn(μ)}y^{mc_{t}(μ)}z^{mc_{w}(μ)}  (15)

= ∑_{μ} log (\frac{1}{1 − x^{n(μ)}y^{c_{t}(μ)}z^{c_{w}(μ)}} ) .  (16)

Similarly, let λ_{F,n,c_{t},c_{w}} be the total number of F-wild partitions of degree n, tame conductor c_{t}, and wild conductor c_{w}. Then its generating function is expressible as a product over irreducibles,

Λ_{F}(x, y, z) = ∑_{n=0}^{∞} ∑_{c_{t}=0}^{∞} ∑_{c_{w}=0}^{∞} λ_{F,n,c_{t},c_{w}}x^{n}y^{c_{t}}z^{c_{w}}  (17)

= ∏_{μ} ∑_{m=0}^{∞} x^{mn(μ)}y^{mc_{t}(μ)}z^{mc_{w}(μ)}  (18)

= ∏_{μ} (\frac{1}{1 − x^{n(μ)}y^{c_{t}(μ)}z^{c_{w}(μ)}} ) .  (19)

The presence of 1/m in (15) and the absence of a corresponding factor in (18) reflects that (14)-(16) are in the setting of total mass while (17)-(19) are in the setting of total number.

Comparison of (14)-(16) with (17)-(19) shows that one has an exponential formula

Λ_{F}(x, y, z) = exp (Φ_{F}(x, y, z)) .  (20)

We will have analogous exponential formulas on the level of fields and algebras in the sequel.

Computation of Φ_{F}(x, y, z) and Λ_{F}(x, y, z). The degree n of an isotypical partition factors into ef, where e is the degree of any constituent part and f is the number of such parts. Following the terminology of the number-theoretic situation from which we are abstracting, we call e the ramification index and f the inertial degree. In our continuing example, both (3_{1,i} + 3_{1,−i}) and (2 + 2 + 2) have degree n = 6. The corresponding (e, f) are (3, 2) in the first case and (2, 3) in the second. The tame conductor is always c_{t} = (e − 1)f, thus 4 in the first case and 3 in the second. The Swan conductor s of an isotypical partition is the first subscript on any of the parts, these first subscripts being all equal. One has c_{w} = fs, this equation being 2 = 2 · 1 in the first case and 0 = 3 · 0 in the second.

Let ϕ_{F}(e, f, s) be the total mass of isotypical F-wild partitions with ramification index e, inertial degree f, and Swan conductor s. One has

ϕ_{F}(e, f, s) = \frac{1}{f} |W (p, e_{0}, e, s)^{σ^{f}} | = \frac{1}{f} q^{f d(p,e_{0},e,s)} (1 − δ_{s}^{wE}q^{−f}) .  (21)

Here, as usual, X^{g} is the set of fixed points of an operator g on a set X. Also, to unify cases, δ_{s}^{wE} is 1 in the non-maximal case s < wE and 0 in the maximal case s = wE.

Since (21) plays a particularly important role in this paper, we explain how it looks in our continuing example. Take (p, e_{0}, e, s) = (3, 2, 3, 1) so that σ = σ_{3} and take also f = 2. The fixed point set of σ^{2} on W (p, e_{0}, e, s) ≅ \overline{𝔽}_{3} − \{0\} is 𝔽_{9} − \{0\}. The Frobenius operator σ on 𝔽_{9} − {0} has three orbits of size two, corresponding to the irreducible wild partitions 3_{1,ω} + 3_{1,ω^{3}} for ω ∈ {i, 1 + i, 2 + i}. Similarly, σ has two orbits of size one, corresponding to the isotypical but non-irreducible F-wild partitions 3_{1,ω} +3_{1,ω} for ω ∈ {1, 2}; the multiplicity of these isotypical partitions is 2, so each contributes only 1/2 towards the total mass. So in this case the three quantities equated in (21) are all 4.

The concepts of ramification index and inertial degree together with the key formula (21) let us evaluate our first generating function explicitly:

Φ_{F}(x, y, z) = ∑_{e=1}^{∞} ∑_{s∈Ore(p,e_{0},e)} ∑_{f=1}^{∞} ϕ_{F}(e, f, s)x^{ef}y^{(e−1)f}z^{sf}  (22)

= ∑_{e=1}^{∞} ∑_{s∈Ore(p,e_{0},e)} ∑_{f=1}^{∞} \frac{1}{f} q^{d(p,e_{0},e,s)f} (1 − δ_{s}^{wE}q^{−f}) x^{ef}y^{(e−1)f}z^{sf}  (23)

= ∑_{e=1}^{∞} ∑_{s∈Ore(p,e_{0},e)} log (\frac{1 − δ_{s}^{wE}q^{d(p,e_{0},e,s)−1}x^{e}y^{e−1}z^{s}}{1 − q^{d(p,e_{0},e,s)}x^{e}y^{e−1}z^{s}} ) .  (24)

The logarithm appearing in (24) is very welcome because it cancels the exponential in (20) to yield the main result of this section:

Λ_{F}(x, y, z) = ∏_{e=1}^{∞} ∏_{s∈Ore(p,e_{0},e)} (\frac{1 − δ_{s}^{wE}q^{d(p,e_{0},e,s)−1}x^{e}y^{e−1}z^{s}}{1 − q^{d(p,e_{0},e,s)}x^{e}y^{e−1}z^{s}} ) .  (25)

4 The generating function Λ_{Q}(x)

Formula (25) is somewhat complicated, as it involves a product over the set Ore(p, e_{0}, e) and makes reference to the numbers d(p, e_{0}, e, s). In this section, we evaluate the two specializations mentioned in the introduction. In both cases, the set Ore(p, e_{0}, e) and the numbers d(p, e_{0}, e, s) each disappear from the final formula.

The specialization (y, z) = (1, 1/q). Equality (2), namely Λ_{p,e_{0},f_{0}}(x, 1, 1/q) = Λ(x), is established by reducing the e-factor of Λ_{p,e_{0},f_{0}}(x, 1, 1/q) to the e-factor of Λ(x):

Λ_{p,e_{0},f_{0};e}(x, 1, \frac{1}{q}) = (  ∏_{s∈Ore(p,e_{0},e)−\{wE\}} \frac{1 − q^{d(p,e_{0},e,s)−1−s}x^{e}}{1 − q^{d(p,e_{0},e,s)−s}x^{e}} ) \frac{1}{1 − q^{d(p,e_{0},e,wE)−wE}x^{e}}

= (  ∏_{k=0}^{wE−d(p,e_{0},e,wE)−1} \frac{1 − q^{−k−1}x^{e}}{1 − q^{−k}x^{e}} ) \frac{1}{1 − q^{d(p,e_{0},e,wE)−wE}x^{e}}

= \frac{1}{1 − x^{e}} .

Here the first equality is simply a specialization of (25). The second equality holds because as s increases through Ore(p, e_{0}, e), the new index k = s−d(p, e_{0}, e, s) increases by uniform steps of 1 from 0 to wE − d(p, e_{0}, e, wE), as at each step (s, d(p, e_{0}, e, s)) increases by either (1, 0) or (2, 1). The third equality holds because each numerator cancels the next denominator, leaving only the denominator of the initial k = 0 factor.

The specialization (y, z) = (1, 1/q) in the context of p-adic fields and algebras corresponds to counting fields and algebras, but weighting wildly ramified algebras less, in accordance with their wild conductor. In the context of fields, this method of weighting was first introduced by Serre [23]. Serre’s mass formula was translated into the context of algebras by Bhargava [4]. The technique of generating functions was first used in this context by Kedlaya [13]. The specialization (y, z) = (1, 1/q) is the relevant one for the application to number fields made by Bhargava [4], [3]. However to support our Conjecture 11.1 below, we need another specialization.

Our specialization. Our specialization is (y, z) = (1, 1). In fact, if we did not take the detour to evaluate Serre’s specialization, it would have sufficed to take z = y throughout this paper and work with two-variable generating functions. Following the model provided by Serre’s case, we evaluate Λ_{p,e_{0},f_{0}}(x, 1, 1) by evaluating each e-factor separately. As for Serre’s case, the main point is that Λ_{p,e_{0},f_{0};e}(x, 1, 1) is given by a telescoping product. In our case, however, the cancellation is not as complete.

Theorem 4.1. The quantity Λ_{p,e_{0},f_{0};e}(x, 1, 1) depends only on Q = p^{e_{0}f_{0}} and is equal to

Λ_{Q;e}(x) = \frac{∏_{j=0}^{w−1}(1 − Q^{(p^{w}−p^{w−j})t/(p−1)}x^{e})^{(p−1)p^{j}}}{(1 − Q^{(p^{w}−1)t/(p−1)}x^{e})^{p^{w}}} .  (26)

Proof. Let D(k) = d(p, e_{0}, e, kE − 1) be the number of integers in {0, …, kE − 1} which are not in Ore(p, e_{0}, e). This quantity is best understood by thinking in terms of the j^{th} shifted block [(j − 1)E, jE − 1] rather that the j^{th} block [(j − 1)E + 1, jE]. In these terms, D(k) is the number of omitted entries in the first k shifted blocks of the corresponding Ore table. The j^{th} block contains exactly those integers in [(j − 1)E, jE − 1] which are not multiples of p^{j}. So

D(k) = E ∑_{j=1}^{k} p^{−j} = E \frac{p^{−1} − p^{−k−1}}{1 − p^{−1}} = e_{0}tp^{w} \frac{1 − p^{−k}}{p − 1} = e_{0}t\frac{p^{w} − p^{w−k}}{p − 1} .  (27)

The e-factor Λ_{p,e_{0},f_{0};e}(x, 1, 1) then simplifies to the e-factor of Λ_{Q}(x) as follows:

Λ_{p,e_{0},f_{0};e}(x, 1, 1) = (  ∏_{s∈Ore(p,e_{0},e)−\{wE\}} \frac{1 − q^{d(p,e_{0},e,s)−1}x^{e}}{1 − q^{d(p,e_{0},e,s)}x^{e}} ) \frac{1}{1 − q^{d(p,e_{0},e,wE)}x^{e}}

= (∏_{j=1}^{w} ∏_{d=D(j−1)+1}^{D(j)} (\frac{1 − q^{d−1}x^{e}}{1 − q^{d}x^{e}} )^{p^{j}−1}) \frac{1}{1 − q^{D(w)}x^{e}}

= (∏_{j=1}^{w} (\frac{1 − q^{D(j−1)}x^{e}}{1 − q^{D(j)}x^{e}} )^{p^{j}−1}) \frac{1}{1 − q^{D(w)}x^{e}}

= \frac{∏_{j=1}^{w}(1 − q^{D(j−1)}x^{e})^{(p−1)p^{j−1}}}{(1 − q^{D(w)}x^{e})^{p^{w}}}
= \frac{∏_{j=1}^{w}(1 − Q^{(p^{w}−p^{w−j+1})t/(p−1)}x^{e})^{(p−1)p^{j−1}}}{(1 − Q^{(p^{w}−1)t/(p−1)}x^{e})^{p^{w}}} .

Here the first equality is simply a specialization of (25). The second equality holds because as s increases through Ore(p, e_{0}, e), the quantity d = d(p, e_{0}, e, s) increases from 1 to D(w) by steps of 0 or 1. If d occurs in the k^{th} shifted block, then it occurs exactly p^{k} − 1 times, yielding the same factor each time. The third equality makes cancellations within a shifted block of Ore numbers. The fourth equality makes cancellations between adjacent shifted blocks. The fifth equality uses (27) and the fact that q^{e_{0}} = Q. The statement of the theorem is then obtained by the index shift j ↦ j − 1. (The indexing in the statement is preferable in the sequel; otherwise, e.g., the three p^{j} in (29) would all have to be p^{j−1}.)

For wildness w in {0, 1, 2, 3}, Formula (26) simplifies to

Λ_{Q;t}(x) = \frac{1}{1 − x},
Λ_{Q;pt}(x) = \frac{(1 − x^{e})^{p−1}}{(1 − Q^{t}x^{e})^{p}} ,
Λ_{Q;p^{2}t}(x) = \frac{(1 − x^{e})^{p−1}(1 − Q^{pt}x^{e})^{p^{2}−p}}{(1 − Q^{(p+1)t}x^{e})^{p^{2}}} ,
Λ_{Q;p^{3}t}(x) = \frac{(1 − x^{e})^{p−1}(1 − Q^{p^{2}t}x^{e})^{p^{2}−p}(1 − Q^{(p^{2}+p)t}x^{e})^{p^{3}−p^{2}}}{(1 − Q^{(p^{2}+p+1)t}x^{e})^{p^{3}}} .

These cases more than suffice for the factors of Λ_{2}(x) and Λ_{3}(x) displayed in the introduction. In general, to pass from Λ_{p;e}(x) to Λ_{p^{n_{0}};e}(x), each written in this form, one simply replaces each factor of the form 1 − cx^{e} by a new factor 1 − c^{n_{0}}x^{e}.

For Q regarded as a formal variable, the right side of (26) defines an element of ℤ[Q][[x]], i.e. a formal power series in x with coefficients in the polynomial ring ℤ[Q]. Accordingly, we could allow Q to be any complex number. To not stray too far from our main focus, we allow Q only to be real number ≥ 1, so that Q = p^{ν} for some ν ≥ 0. As remarked already in the introduction, Q no longer determines p in our enlarged context. Accordingly, we write Λ_{p,Q}(x) rather than Λ_{p}(x).

The new flexibility gained by allowing Q to vary offers new insights into our situation. To start, note that if Q = 1 then all factors on the right side of (26) reduce to 1 − x^{e}, with p^{w} − 1 such factors on the top and p^{w} on the bottom. Thus (26) reduces to 1/(1 − x^{e}) independent of the wildness w, and so Λ_{p,1}(x) = Λ(x) for all p. This observation further justifies our terminology “wild partitions,” because wild partitions are now continuously related to ordinary partitions.

5 Relation with p-cores

In this section and the next, we continue with the set-up just introduced, so that Q ≥ 1 is a real number independent from p. Our results in these sections go through even when p is allowed to be any integer larger than one, as long as one still defines wildness by the equation e = tp^{w} and the condition that p does not divide t. However to not superfluously complicate our exposition, we will stick with our requirement that p is prime.

p-cores. Define

Θ_{p}(x) = \frac{Λ(x)}{Λ(x^{p})^{p}} = ( ∏_{ord_{p}(e)≥1}(1 − x^{e})^{p−1}) ( ∏_{ord_{p}(e)=0} \frac{1}{1 − x^{e}} ) .  (28)

The series Θ_{p}(x) is the generating function for the sequence of p-cores, i.e. partitions without hook-lengths a multiple of p, as studied in [6]. In the cases p = 2 and 3, respectively, Θ_{p}(x) is the generating function for \underline{A010054} and \underline{A033687}:

Θ_{2}(x) = ∑_{j=−∞}^{∞} x^{2j^{2}−j}  = 1 + x + x^{3} + x^{6} + x^{10} + x^{15} + x^{21} + x^{28} + ⋯

Θ_{3}(x) = ∑_{j=−∞}^{∞} ∑_{k=−∞}^{∞} x^{3(j^{2}+jk+k^{2})−j−2k} = 1 + x + 2x^{2} + 2x^{4} + x^{5} + 2x^{6} + x^{8} + 2x^{9} + ⋯

The reference [6] similarly identifies Θ_{p}(x) as a theta-series of a quadratic form on ℤ^{p−1}. By either the combinatoric interpretation or the theta-series interpretation, one gets that Θ_{p}(x) has non-negative coefficients, something not obvious from (28).

The analytic properties of Θ_{p}(x) are relevant for the asymptotics of the next section. Figure 1 provides a guide. For all p, the radius of convergence is 1. In fact, the sum of the first n coefficients of Θ_{p}(x)^{k} is approximately the volume of a ball of radius ρ = \sqrt{2n}p^{1/2+1/(2p−2)} in Euclidean space of dimension m = (p − 1)k, thus π^{m/2}ρ^{m}/(m/2)!. Because all coefficients are non-negative, for fixed 0 < r < 1 the function |Θ_{p}(re^{iθ})| takes on its unique maximum at θ = 0.

Λ_{p,Q}(x) as a product over p-cores.

Figure 1: Contour plots of |Θ_{2}(x)| on the left and |Θ_{3}(x)| on the right, both on the open unit disk |x| < 1, with dark shading indicating large values. The arguments indicated by dots are the ones relevant for Table 1. They give the main contribution to the numbers in Table 2.

Corollary 5.1. One has the identity

Λ_{p,Q}(x) = ∏_{j=0}^{∞} Θ_{p}(Q^{(p^{j}−1)/(p−1)}x^{p^{j}})^{p^{j}}.  (29)

Before beginning the general proof, it is worth noting that the right side of (29) in the case of Q = 1 is a telescoping product, namely

Θ_{p}(x)Θ_{p}(x^{p})^{p}Θ_{p}(x^{p^{2}})^{p^{2}} ⋯ = \frac{Λ(x)}{Λ(x^{p})^{p}} \frac{Λ(x^{p})^{p}}{Λ(x^{p^{2}})^{p^{2}}} \frac{Λ(x^{p^{2}})^{p^{2}}}{Λ(x^{p^{3}})^{p^{3}}} ⋯ = Λ(x).  (30)

This remark establishes (29) in the case Q = 1, as we have already remarked at the end of the previous section that Λ_{p,1}(x) = Λ(x).

Proof. Substituting Q^{(p^{j}−1)/(p−1)}x^{p^{j}} in for x in (28), we get that the factor on the right of (29) with index j is

Θ_{p}(Q^{(p^{j}−1)/(p−1)}x^{p^{j}})^{p^{j}}

= ( ∏_{ord_{p}(e)>0}(1 − Q^{(p^{j}−1)e/(p−1)}x^{p^{j}e}))^{p^{j+1}−p^{j}} ( ∏_{ord_{p}(e)=0} \frac{1}{1 − Q^{(p^{j}−1)e/(p−1)}x^{p^{j}e}} )^{p^{j}}

= ( ∏_{ord_{p}(e)>j}(1 − Q^{(1−p^{−j})e/(p−1)}x^{e}))^{p^{j+1}−p^{j}} ( ∏_{ord_{p}(e)=j} \frac{1}{1 − Q^{(1−p^{−j})e/(p−1)}x^{e}} )^{p^{j}} (31)

Let e = p^{w}t as usual. Then the e-factor of the left side of (29) is given by (26) and the e-factor of the right side of (29) is the product of (31) for j = 0, …, w. The w numerator factors of (26) match the numerator factors in (31) for j = 0, …, w − 1. The denominator factor of (26) matches the denominator factor in (31) for j = w.

6 Asymptotics

In this section we study the asymptotic behavior of the coefficients λ_{p,Q,n} of the power series Λ_{p,Q}(x).

A change of variables. Abbreviate Q^{−1/(p−1)} by r ∈ (0, 1] and change variables via x = ry. Define \underline{Λ}_{p,Q}(y) = Λ_{p,Q}(ry) = ∑_{n=0}^{∞} \underline{λ}_{p,Q,n}y^{n} so that \underline{λ}_{p,Q,n} = r^{n}λ_{p,Q,n}. Then (29) takes on the simpler form

\underline{Λ}_{p,Q}(y) = ∏_{j=0}^{∞} Θ_{p}(ry^{p^{j}})^{p^{j}}.  (32)

As the radius of convergence of Θ_{p}(x) is 1, the radius of convergence of the j^{th} factor is r^{−1/p^{j}}. So as j increases from 1, these radii decrease monotonically from 1/r with limit 1.

Figure 2: Points (n, log(\underline{λ}_{2,2,n;k})) on the left and (n, log(\underline{λ}_{3,3,n;k})) on the right, with k indicated by text.

We indicate with a subscript k the corresponding objects where the product in (32) is taken from 0 to k. For any given k, the coefficients \underline{λ}_{p,Q,n;k} eventually decay exponentially, by the above radius of convergence remarks. Figure 2 illustrates the decay.

Radius of convergence and upper bounds for λ_{p,Q,n}. We are interested more in the behavior of the coefficients \underline{λ}_{p,Q,n} themselves, rather than the cutoff versions \underline{λ}_{p,Q,n;k}. In general, let Θ(z) be any power series convergent at least on the closed disk of radius r with Θ(0) = 1 and |Θ(r)| > 1. Then it is elementary that ∏_{j=0}^{∞} Θ(ry^{p^{j}})^{p^{j}} converges for |y| < 1. The product does not converge for y = 1 because it formally has the form Θ(r)^{∞}. Thus our \underline{Λ}_{p,Q}(y) have radius of convergence exactly one. In terms of Figure 2, the upper envelope of the points plotted grows at most sub-linearly. In terms of the original λ_{p,Q,n}, one eventually has λ_{p,Q,n} < Q^{n/(p−1)+ϵ} for any positive ϵ.

Root growth. Sharpening the statement lim sup λ_{p,Q,n}^{1/n} = Q^{1/(p−1)} just observed is the following.

Proposition 6.1. lim_{n→∞} λ_{p,Q,n}^{1/n} = Q^{1/(p−1)}.

Proof. We need only show that the sequence λ_{p,Q,n}^{1/n} has no limit points smaller than Q^{1/(p−1)}. In other words, we need only show that the sequence \underline{λ}_{p,Q,n}^{1/n} has no limit points smaller than 1. From (28), we know that the expansion of Θ_{p}(x) begins with ∑_{n=0}^{p−1} λ_{n}x^{n} which is termby-term at least ∑_{n=0}^{p−1} x^{n}. The j^{th} factor Θ_{p}(ry^{p})^{p^{j}} of (29) is coefficient-wise bounded from below by Θ_{p}(ry^{p}) which is in turn coefficient-wise bounded from below by ∑_{a=0}^{p−1} r^{a}y^{ap^{j}}. So Λ_{p,Q}(y) is coefficient-wise bounded from below by ∏_{j=0}^{∞} ∑_{a=0}^{p−1} r^{a}y^{ap^{j}}. If n = ∑_{i=0}^{log_{p}(n)} a_{i}p^{i} with 0 ≤ a_{i} ≤ p − 1, then

\underline{λ}_{p,Q,n} ≥ ∏_{i=0}^{log_{p}(n)} r^{a_{i}} ≥ r^{(1+log_{p}(n))(p−1)}

so that

\underline{λ}_{p,Q,n}^{1/n} ≥ r^{(1+log_{p}(n))(p−1)/n}.  (33)

The right side of (33) tends to 1 with n, proving that indeed \underline{λ}_{p,Q,n}^{1/n} has no limit points smaller than 1.

Expected refined asymptotics. Proposition 6.1 is more than we need to support Conjecture 11.1 below. However the extreme crudeness of the bounds in its proof suggests that stronger statements are provable. Rather than proceed incrementally, in the rest of this section we present numerical evidence and heuristic argument leading up to the very strong statement (37), conjecturally extending the Hardy-Ramanujan asymptotic for λ_{n}.

Another model of the type of statement sought is given in the next section for analogous quantities λ_{F,n} for F = ℝ and F = ℂ. There, (42) gives an asymptotic equivalent to the root decay factor λ_{F,n}^{1/n}, analogous to our Proposition 6.1. More subtly, (42) also says that the ratio decay factor λ_{F,n}/λ_{F,n−1} has the same asymptotic equivalent. Finally (41) is the sharpest statement, giving asymptotic equivalents to λ_{F,n} itself.

Ratio oscillation. In contrast to both ordinary partitions and the Archimedean cases, evidence strongly suggests that λ_{p,Q,n}/λ_{p,Q,n−1} does not have a limiting value for Q > 1. Figure 3 graphs log(\underline{λ}_{p,Q,n}) for p = 2, 3 and Q = p^{j} with j = 1, 2. One sees smooth growth Figure 3: Points (n, log(\underline{λ}_{2,2,n})) on the left and (n, log(\underline{λ}_{3,3,n})) on the right, for 0 ≤ n ≤ 100. Also points (n, log(\underline{λ}_{2,4,n})) on the left and (n, log(\underline{λ}_{3,9,n})) on the right, for 36 ≤ n ≤ 100. The oscillatory behavior modulo eight on the left and modulo nine on the right matches Table 2 well.

with oscillatory behavior superimposed. The corresponding picture for n out through 4000 shows no damping. In general, oscillatory behavior is barely visible for Q near 1 and increases in amplitude with Q. There seem to be dominant oscillations with period p, secondary oscillations with period p^{2}, tertiary oscillations with period p^{3}, and so on. The situation clearly calls for a Fourier analysis.

Figure 4 illustrates with two examples the magnitude of \underline{Λ}_{p,Q}(y) as a function of the complex variable y. The needed Fourier analysis is connected with the limiting behavior of \underline{Λ}_{p,Q}(y) as |y| increases to 1. For r < 1, consider the function

ĉ_{p,Q}(y) = ∏_{j=0}^{∞} (\frac{Θ_{p}(ry^{p^{j}})}{Θ_{p}(r)})^{p^{j}}  (34)

defined on the unit circle. One has ĉ_{p,Q}(1) = 1 as indeed all factors are 1. For y satisfying y^{p^{k}} = 1 but not y^{p^{k−1}} = 1 the infinite product reduces to the product of its first k factors, all of which are non-zero with absolute value less than one; thus 0 < |ĉ_{p,Q}(y)| < 1 for these y. Finally if y is otherwise, the infinite product converges to 0.

We view ĉ_{p,Q}(y) as giving the normalized boundary values of \underline{Λ}_{p,Q}(y). Intuitively, we can view \underline{Λ}_{p,Q}(y) as having its most important singularity at 1. This singularity is echoed in quantitatively smaller singularities at primitive p^{th} roots of unity. It is echoed in still smaller singularities at primitive roots of unity of order p^{2}, and so on, as illustrated by Figure 4. Table 1 gives a more numerical illustration, and includes also the cases (p, Q) = (2, 4) and (p, Q) = (3, 9). For given p, the echoes decay more slowly with larger Q.

Figure 4: Contour plots of |\underline{Λ}_{2,2}(y)| on the left and |Λ_{3,3}(y)| on the right, both on the open unit disk |y| < 1, with dark shading indicating large values.

The function ĉ_{p,Q}(y) enters into our Fourier analysis as follows. Taking an inverse Fourier transform, define

c_{p,Q}(n) = ∑_{y} y^{−n}ĉ_{p,Q}(y),  (35)

the sum being over all p^{th} power roots of unity. One can check that the sum in (35) indeed converges. Moreover, let ℤ_{p} be the p-adic integers, i.e. the completion of ℤ with respect to the sequence of finite quotients ℤ/p^{j}ℤ. Then c_{p,Q}, thought of as a function from ℤ to ℝ, extends continuously to a function from ℤ_{p} to ℝ. Table 2 gives some values. We expect that

\begin{matrix}α & ĉ_{2,2}(e^{2πiα}) & ĉ_{2,4}(e^{2πiα}) & α & ĉ_{3,3}(e^{2πiα}) & ĉ_{3,9}(e^{2πiα}) \\ 0/1 & 1. & 1. & 0/1 & 1. & 1. \\ 1/8 & 0.0005 + 0.0009i & 0.0567 + 0.0405i & 1/9 & 0.0004 + 0.0010i & 0.0542 + 0.0569i \\ 1/4 & 0.0341 + 0.0130i & 0.2660 + 0.0624i & 2/9 & 0.0006 + 0.0002i & 0.0560 − 0.0029i \\ 3/8 & 0.0007 − 0.0001i & 0.0490 − 0.0119i & 1/3 & 0.1191 + 0.0210i & 0.4476 + 0.0718i \\ 1/2 & 0.2385 & 0.5803 & 4/9 & 0.0007 + 0.0001i & 0.0463 + 0.0205i \end{matrix}

Table 1: Some normalized boundary values ĉ_{p,Q}(y) of \underline{Λ}_{p,Q}(y), rounded to the nearest ten-thousandth. Only values in the upper half plane are given, because ĉ_{p,Q}(ȳ) = \overline{ĉ_{p,Q}(y)}.

n c_{2,2}(n) c_{2,4}(n)  n c_{3,3}(n) c_{3,9}(n)
0 1.309 2.324  0 1.242 2.208
1 0.788 0.596  1 0.918 0.774
2 1.172 1.153  2 0.847 0.496
3 0.737 0.324  3 1.238 1.878
4 1.304 1.901  4 0.918 0.679
5 0.787 0.493  5 0.845 0.426
6 1.168 0.944  6 1.235 1.600
7 0.734 0.265  7 0.915 0.577
8 0.842 0.362

Table 2: Some values of c_{p,Q}(n), rounded to the nearest thousandth. To the nearest thousandth, c_{2,2}(n) depends only on n modulo 8 while c_{3,3}(n) depends only on n modulo 9. Similarly, to the nearest hundredth, c_{2,4}(n) and c_{3,9}(n) depend only on n to the respective moduli 8 and 9.

the function c_{p,Q}(n) fully captures the oscillatory behavior in the sense that

\frac{\underline{λ}_{p,Q,n}/c_{p,Q}(n)}{\underline{λ}_{p,Q,n−1}/c_{p,Q}(n − 1)} ∼ Q^{1/(p−1)}.  (36)

Computations such as those illustrated by Figure 5 support this expectation.

Towards an asymptotic equivalent to λ_{p,Q,n}. The smooth part of λ_{p,Q,n} presents more of a mystery. Computations are consistent with the conjecture given as (5) in the introduction, namely

λ_{p,Q,n} ∼ c_{p,Q}(n)C_{p}(Q)n^{B_{p}(Q)}e^{A_{p}(Q)\sqrt{n}}Q^{n/(p−1)}  (37)

for quantities A_{p}(Q), B_{p}(Q), and C_{p}(Q) to be thought of as functions on the Q-interval [1, ∞).

For Q = 1, the oscillatory factor c_{p,Q}(n) reduces to 1. One has

(A_{p}(1), B_{p}(1), C_{p}(1)) = (π\sqrt{\frac{2}{3}}, −1, \frac{1}{4\sqrt{3}}) ≈ (2.56, −1.00, 0.144),

independently of p, by the Hardy-Ramanujan asymptotic for partitions [8]. Figure 5 graphs functions A_{p}(Q)\sqrt{n} + B_{p}(Q) log n + log C_{p}(Q) with (A_{p}(Q), B_{p}(Q), C_{p}(Q)) deduced from a least squares fit to log(\underline{λ}_{p,Q,n}/c_{p,Q}(n)) over [20, 4000]. The drawn lines are thick enough so that they contain all the actual points (n, log(\underline{λ}_{p,Q,n}/c_{p,Q}(n))). For the most important quantity A_{p}(Q), the fit yields

A_{2}(2) ≈ 1.66 ,  A_{3}(3) ≈ 1.68 ,
A_{2}(4) ≈ 1.18 ,  A_{3}(9) ≈ 1.21 .

Numeric computations are not accurate enough to suggest an analytic form for A_{p}(Q), B_{p}(Q), and C_{p}(Q); a more theoretical approach is needed.

Figure 5: Least square fits to points (n, log(\underline{λ}_{2,2^{j},n}/c_{2,2^{j}}(n))) on the left and (n, log(\underline{λ}_{3,3^{j},n}/c_{3,3^{j}}(n))) on the right, for j ∈ \{1, 2\} and 20 ≤ n ≤ 4000.

7 Fields, algebras, and the cases F = ℝ and F = ℂ

In this section, we introduce some concepts associated to field and algebra extensions of a given ground field F. These concepts will play a major role in the rest of the paper. Also we illustrate these concepts with the ground fields F = ℝ and F = ℂ. These ground fields are particularly simple and familiar. Moreover, they play an essential role in the global considerations of Section 11.

Fields and algebras. For F a field, let Fields_{F,n} be the set of isomorphism classes of separable degree n field extensions of F. Our main interest is in characteristic zero, where all fields are separable; accordingly we drop the adjective “separable.” Similarly, let Algebras_{F,n} be the set of isomorphism classes of degree n algebra extensions which are products of field extensions. For both fields and algebras, we allow ourselves also to drop the phrase “of isomorphism classes” since it always understood. Similarly we write just K instead of say [K] to indicate the isomorphism class of an algebra K.

An algebra K has an automorphism group Aut(K/F). We define, as is standard, its mass to be 1/|Aut(K/F)|. For some fields F, all the sets Fields_{F,n} are finite. Exactly in this case, all the larger sets Algebras_{F,n} are finite too. For these F, we define ϕ_{F,n} and λ_{F,n} to be the total masses of Fields_{F,n} and Algebras_{F,n} respectively. Let

Φ_{F} (x) = ∑_{n=1}^{∞} ϕ_{F,n}x^{n} = x + ⋯,  Λ_{F} (x) = ∑_{n=0}^{∞} λ_{F,n}x^{n} = 1 + x + ⋯

be the corresponding generating functions. Then one has the exponential formula [25, Chapter 5]

Λ_{F} (x) = exp(Φ_{F} (x)),  (38)

from the definition of mass and the way algebras are built from fields.

The cases F = ℝ and F = ℂ. With the above definitions, Fields_{ℝ,1} = {ℝ}, Fields_{ℝ,2} = {ℂ}, and otherwise Fields_{ℝ,n} = ∅. Also Algebras_{ℝ,n} = {ℝ^{r}ℂ^{s} : r + 2s = n}, with the mass of ℝ^{r}ℂ^{s} being 1/(r!s!2^{s}). Even more simply, the only non-empty Fields_{ℂ,n} is Fields_{ℂ,1} = {ℂ}. One has Algebras_{ℂ,n} = {ℂ^{n}}, with the mass of ℂ^{n} being 1/n!. Thus

Λ_{ℝ}(x) = ∑_{n=0}^{∞} λ_{ℝ,n}x^{n} = e^{x+x^{2}/2} = 1 + x + \frac{2}{2}x^{2} + \frac{4}{6}x^{3} + \frac{10}{24}x^{4} + \frac{26}{120}x^{5} + ⋯, (39)

Λ_{ℂ}(x) = ∑_{n=0}^{∞} λ_{ℂ,n}x^{n} = e^{x}  = 1 + x + \frac{1}{2}x^{2} + \frac{1}{6}x^{3} + \frac{1}{24}x^{4} + \frac{1}{120}x^{5} + · · · . (40)

The numbers n!λ_{ℝ,n} form the sequence \underline{A000085} giving, among other interpretations, the number of involutions in the symmetric group S_{n}. One has the asymptotic formulas

λ_{ℝ,n} ∼ \frac{e^{\frac{n}{2} +\sqrt{n}− \frac{1}{4}} n^{− \frac{n}{2} − \frac{1}{2}}}{2\sqrt{π}} ,  λ_{ℂ,n} ∼ \frac{e^{n}}{n^{n+ \frac{1}{2}} \sqrt{2π}} ,  (41)

the first due to Moser and Wyman [17] and the second being Stirling’s approximation. On the level of ratio and root behavior, one has

\frac{λ_{ℝ,n}}{λ_{ℝ,n−1}} ∼ λ_{ℝ,n}^{1/n} ∼ \sqrt{\frac{e}{n}},  \frac{λ_{ℂ,n}}{λ_{ℂ,n−1}} ∼ λ_{ℂ,n}^{1/n} ∼ \frac{e}{n},  (42)

thus superexponential decay.

8 Eisenstein polynomials

For this and the next two sections, fix a prime number p. Let ℚ_{p} be the field of p-adic numbers. Its ring of integers ℤ_{p} already arose naturally in Section 6. The maximal ideal of ℤ_{p} is generated by the prime number p, and the corresponding residue field is 𝔽_{p} = ℤ_{p}/p. For background on p-adic numbers, see e.g. [7]. We need mainly the algebraic theory of finite degree field extensions of ℚ_{p}, i.e. Chapter 5 of [7].

The ground field F. For this and the next two sections, fix also an extension field F of degree n_{0} over ℚ_{p}. So F can be presented as ℚ_{p}[x]/g_{0}(x) for some irreducible polynomial g_{0}(x) in ℚ_{p}[x] of degree n_{0}. We have no need to consider g_{0}(x) again, as we will simply regard F as given. Let O be the ring of integers of F, let Π be its maximal ideal, and let κ = O/Π. The ramification index of F/ℚ_{p} is the positive integer e_{0} such that Π^{e_{0}} = (p). The inertial degree of F/ℚ_{p} is the positive integer f_{0} such that q := |κ| = p^{f_{0}}. One has e_{0}f_{0} = n_{0}.

It is often clearer to avoid the language of ideals. To do this we fix a uniformizer π of F, i.e. a generator of Π. For a ∈ O − {0}, we write ord_{π}(a) = b to mean that a generates the ideal (π^{b}). We define ord_{π} on all of O by writing ord_{π}(0) = ∞.

Extensions of F and their numerical invariants. Likewise, one can consider field extensions K = F [x]/g(x) of F. The degree n of such an extension factors into its ramification index e and its inertial degree f. Another important invariant of a field extension K/F is its discriminant d(K/F), which is an ideal Π^{c} in O. We focus on the discriminant-exponent c, which we call the conductor. If K = F [x]/g(x), then the ideal generated by the polynomial discriminant

D(g) = (−1)^{n(n−1)/2}Res_{x}(g(x), g′(x))  (43)

has the form Π^{c+2d} for d a non-negative integer, called the defect of g(x).

The conductor of K/F is naturally written as c = c_{t} + c_{w}, where c_{t} is the tame conductor and c_{w} is the wild conductor. Very simply, c_{t} = f(e − 1). The wild conductor c_{w} is more complicated, but has the form fs, where s ∈ Ore(p, e_{0}, e) is a non-negative integer called the Swan conductor, as detailed below.

We seek to understand the sets Fields_{F,n} introduced in the previous section. The decomposition

Fields_{F,n} = ∐_{ef=n} ∐_{s∈Ore(p,e_{0},e)} Fields_{F} (e, f, s)  (44)

is a natural starting point. In the rest of this section, we explain how Eisenstein polynomials give an explicit understanding of the totally ramified part Fields_{F} (e, 1, s). The cases f > 1 are easily reduced to the case f = 1, as explained in the next section.

Eisenstein polynomials. Consider monic polynomials of degree e with coefficients in O. Such a polynomial

g(x) = x^{e} + a_{e−1}x^{e−1} + ⋯ + a_{1}x + a_{0}  (45)

is called an Eisenstein polynomial if and only if π divides all the coefficients a_{i} and moreover π^{2} does not divide a_{0}. Let Eis(O, e) be the space of degree e Eisenstein polynomials over O.

If g(x) is an Eisenstein polynomial then K = F [x]/g(x) is a totally ramified field extension of F. Moreover O[x]/g(x) is its ring of integers which means that the defect d of g(x) is zero. The element x ∈ O[x]/g(x) is a uniformizer, meaning that it generates the maximal ideal of O[x]/g(x).

Conversely, suppose a totally ramified K is given. Then one can consider for each of its uniformizers ω the characteristic polynomial g_{ω}(x) of ω acting by multiplication on K, where K is considered as an e-dimensional vector space over F. The resulting map

ω ↦ g_{ω}(x)  (46)

is |Aut(K/F)|-to-1 over its image Eis(O, e)_{K} ⊆ Eis(O, e).

Conductors of Eisenstein polynomials. The conductor c = c_{t} + c_{w} of the Eisenstein polynomial (45) is ord_{x}(g′(x)), where x here is understood as the given uniformizer of O[x]/g(x) [22, III.6]. The tame conductor c_{t} is e − 1. Thus

c_{w} = ord_{x}(g′(x)) − (e − 1).  (47)

For i = 1, …, e, define the i^{th} index of an Eisenstein polynomial (45) to be

ind_{i}(g(x)) = ord_{x}(ia_{i}x^{i−1}) − (e − 1) = e ord_{π}(ia_{i}) + i − e = \{ \begin{matrix}e ord_{π}(i\frac{a_{i}}{π}) + i, & if i < e, \\ ee_{0}w = wE, & if i = e. \end{matrix}

One has ind_{i}(g(x)) ≡ i modulo e, and so the ind_{i}(g(x)) are all different. The conclusion of these considerations is that the wild conductor c_{w} of g(x) is the smallest of the indices ind_{i}(g(x)).

In words, if i < e then ind_{i}(g(x)) is either greater than wE, and hence irrelevant, or the ord_{π}(a_{i})^{th} number in column i of the corresponding Ore array Ore(p, e_{0}, e). Display (48), a copy of (9) except that Column i has been headed by the corresponding coefficient a_{i}, illustrates this viewpoint.

\begin{matrix}a_{9} & a_{8} & a_{7} & a_{6} & a_{5} & a_{4} & a_{3} & a_{2} & a_{1} \\ . \\ . & 8 & 7 & . & 5 & 4 & . & 2 & 1 \\ . & 17 & 16 & . & 14 & 13 & . & 11 & 10 \\ . & 26 & 25 & 24 & 23 & 22 & 21 & 20 & 19 \\ 36 & 35 & 34 & 33 & 32 & 31 & 30 & 29 & 28 \end{matrix}  (48)

For this displayed case (p, e_{0}, e) = (3, 2, 9), the wild conductor is the number to the right of the first condition that holds:

ord_{π}(a_{1}) = 1  1
ord_{π}(a_{2}) = 1  2
ord_{π}(a_{4}) = 1  4
ord_{π}(a_{5}) = 1  5
⋮  ⋮
ord_{π}(a_{6}) = 2  33
ord_{π}(a_{7}) = 4  34
ord_{π}(a_{8}) = 4  35
ord_{π}(a_{9}) = 0  36

(49)

One has a natural decomposition

Eis(O, e) = ∐_{s∈Ore(p,e_{0},e)} Eis(O, e, s)  (50)
= ∐_{s∈Ore(p,e_{0},e)} ∐_{K∈Fields_{F} (e,1,s)} Eis(O, e, s)_{K}  (51)

where Eis(O, e, s) consists of all Eisenstein polynomials over O of degree e and wild conductor s and Eis(O, e, s)_{K} is the subset consisting of polynomials which define K.

9 p-adic algebras: masses

Let Fields_{F,n,c_{t},c_{w}} be the set of fields over F with degree n, tame conductor c_{t}, and wild conductor c_{w}. Let ϕ_{F,n,c_{t},c_{w}} be the total mass of Fields_{F,n,c_{t},c_{w}}. Let Algebras_{F,n,c_{t},c_{w}} be the corresponding set of algebras and let λ_{F,n,c_{t},c_{w}} be its total mass. One has the corresponding generating functions, satisfying Λ_{F} (x, y, z) = exp(Φ_{F} (x, y, z)).

Let F = (p, e_{0}, f_{0}) be the invariants of F. This section explains how the equality

Φ_{F} (x, y, z) = Φ_{F}(x, y, z)  (52)

follows from the Krasner mass formula. It is this equality which renders the directly defined power series Φ_{F}(x, y, z) of interest in algebraic number theory. By exponentiating both sides of (52) one immediately gets Λ_{F} (x, y, z) = Λ_{F}(x, y, z). Besides our given F = F_{1}, there may be non-isomorphic F_{2}, …, F_{m} with the same invariants (p, e_{0}, f_{0}). Our notation encourages one to also think of F as representing the numerical equivalence class {F_{1}, …, F_{m}}. Note that if p|e_{0} then the different F_{i} have Swan conductors s_{0} varying over Ore(p, 1, e_{0}), but that not even s_{0} enters our considerations.

Volumes. The space of all monic polynomials of degree e with coefficients in O is naturally identified with O^{e} via the coefficients. The quotient space (O/Π^{i})^{e} is a discrete set of size q^{ie}. We view O^{e} as a measure space of mass one by requiring that for all i, each fiber of O^{e} → (O/Π^{i})^{e}, has mass 1/q^{ie}. In this measure, Eis(O, e) clearly has volume q^{−e}(1 − q^{−1}).

As in Section 2, for s ∈ Ore(p, e_{0}, e) let d(p, e_{0}, e, s) be the number of integers in {0, …, s} which are not in Ore(p, e_{0}, e). As illustrated by (49), for a random polynomial to be in Eis(O, e, s) it has to fail s − d(p, e_{0}, e, s) successive tests, each of which is failed with probability 1/q. If s < wE, then it moreover has to pass the next test. Accordingly,

volume(Eis(O, e, s)) = q^{−e−s+d(p,e_{0},e,s)}(1 − q^{−1})(1 − δ_{s}^{wE}q^{−1}),  (53)

with δ_{s}^{wE} either 1 or 0 according to whether s < wE or s = wE, as in Section 3.

The Krasner mass formula and proof of (52). For K ∈ Fields_{F} (e, 1, s), let U_{K} be its set of uniformizers and let Eis(O, e, s)_{K} be the set of its defining Eisenstein polynomials. Consider again the degree |Aut(K/F)| cover U_{K} → Eis(O, e, s)_{K} of (46). By a Jacobian computation [23], one has

mass(K) = q^{s+e} \frac{volume(Eis(O, e, s)_{K})}{1 − q^{−1}}.  (54)

Summing (54) over K ∈ Fields_{F} (e, 1, s) and eliminating volumes via (53) gives

ϕ_{F} (e, 1, s) = q^{d(p,e_{0},e,s)}(1 − δ_{s}^{wE}q^{−1}).  (55)

which is the Krasner mass formula [15].

The case of general f reduces to the totally ramified case f = 1 via

Fields_{F} (e, f, s) = Fields_{F_{f}} (e, 1, s),  (56)

where F_{f} is the unique up to isomorphism degree f extension of F. One has

ϕ_{F} (e, f, s) = \frac{1}{f} ϕ_{F_{f}} (e, 1, s) = \frac{1}{f} q^{f d(p,e_{0},e,s)}(1 − δ_{s}^{wE}q^{−f}).  (57)

Here the first equality of (57) holds because of (56) and the fact that |Aut(F_{f}/F)| = f. The second equality of (57) holds because of (55), with F replaced by F_{f} on the left and hence q replaced by q^{f} on the right.

The quantities ϕ_{F}(e, f, s) and ϕ_{F} (e, f, s) agree because their explicit formulas in (21) and (57) agree. Replacing F by F in the evaluation (22)-(24) of Φ_{F}(x, y, z) then shows that Φ_{F} (x, y, z) evaluates to the same explicit formula.

10 p-adic algebras: geometric packets

One obvious difference between the standard Λ_{ℝ}(x), Λ_{ℂ}(x) and our more complicated Λ_{Q}(x) is that the coefficients of the former decay while the coefficients of the latter grow. Another important difference is that the former have non-integral coefficients while the latter, and even the underlying Λ_{p,e_{0},f_{0}}(x, y, z), have integer coefficients. In this section, we take a new perspective which explains this integrality conceptually. While the previous section justifies our definitions in Section 2 and 3 numerically, this section goes farther and justifies our combinatorial definitions set-theoretically, explaining the natural objects to which wild partitions correspond.

Algebras over F^{un}. For the new perspective, we fix a maximal unramified extension F^{un}/F, with ring of integers O^{un} and associated residual extension \overline{κ}/κ. We let σ ∈ Gal(\overline{κ}/κ) = Gal(F^{un}/F) be the Frobenius element.

The theory of Eisenstein series over O goes through without change over O^{un}. Accordingly, the important set Fields_{F^{un},e} of degree e field extensions of F^{un} is identified with the quotient of Eis(O^{un}, e) modulo an equivalence relation ∼, where g_{1}(x) ∼ g_{2}(x) if and only if F^{un}[x]/g_{1}(x) and F^{un}[x]/g_{2}(x) are isomorphic. One likewise has Fields_{F^{un}}(e, s) = Eis(O^{un}, e, s)/ ∼, these sets being non-empty exactly for s ∈ Ore(p, e_{0}, e). Thus

Fields_{F^{un}} = ∐_{e=1}^{∞} ∐_{s∈Ore(p,e_{0},e)} Fields_{F^{un}}(e, s).  (58)

The Frobenius element σ acts compatibly on both sides of (58) by taking K = F^{un}[x]/ ∑ a_{i}x^{i} to K^{σ} = F^{un}[x]/ ∑ a_{i}^{σ}x^{i}. In turn, as for any ground field, Algebras_{F^{un}} is the free abelian monoid generated by Fields_{F^{un}}. We denote by Algebras_{F^{un}}^{σ} the set of σ-fixed points, as usual.

Geometric packets. The mass formulas of the previous section transfer to cardinality formulas in our new context as follows. If K ∈ Algebras_{F}, then one has its corresponding basechanged algebra K^{un} ∈ Algebras_{F^{un}}^{σ}. Explicitly, if K = F [x]/g(x) then K^{un} = F^{un}[x]/g(x). The fiber of the map

Algebras_{F} → Algebras_{F^{un}}^{σ}  (59)

above a point L ∈ Algebras_{F^{un}}^{σ} is the set of all K ∈ Algebras_{F} with K^{un} ≅ L, i.e. the set of all models of L. These fibers are the geometric packets of the section title. If K_{1} and K_{2} are in the same fiber, one says they are geometrically equivalent.

The main point letting one convert mass formulas to cardinality formulas is that every geometric packet has total mass one. This is a standard fact from descent theory, but we review the proof here because of the critical role it plays for us. Fix L ∈ Algebras_{F^{un}} and let A = Aut(L/F^{un}). Let A = Aut(L/F). Then one has a short-exact sequence

A ↪ A ↠ Gal(\overline{κ}/κ).  (60)

Let A^{1} be the set of preimages of σ in A. So A acts by conjugation on A^{1}. Then each ρ ∈ A^{1} determines a model L^{ρ}/F of L/F^{un}. In fact, L^{ρ} is just the fixed algebra of the subgroup of A generated by ρ. Also elements of A^{1} determine isomorphic models if and only if they differ by conjugation by an element of A; thus the isomorphism class of L^{ρ} is determined by the conjugacy class [ρ] of ρ. Also, the automorphism group of L^{ρ}/F is the subgroup of A which fixes ρ. Finally, one has always

∑_{ρ} \frac{1}{|Aut(L^{ρ}/F)|} = ∑_{ρ} \frac{|[ρ]|}{|A|} = 1,  (61)

each sum being over representatives of the conjugacy classes in A^{1}. The last equality in (61) follows because A^{1} is partitioned into the classes [ρ] and |A| = |A^{1}|.

Wild partitions and F^{un}-algebras. We have established

|W (p, e_{0}, e, s)^{σ^{m}} | = |Fields_{F^{un}}(e, s)^{σ^{m}} |  (62)

as both sides are q^{f d(p,e_{0},e,s)}(1 − δ_{s}^{wE}q^{−f}), the left side via (21) and the right side via (57) and the mass one principle. As all orbits on both sides are finite, this implies

W (p, e_{0}, e, s) ≅ Fields_{F^{un}}(e, s)  (63)

as σ-sets. A choice of bijections (63) induces a bijection from (p, e_{0}, f_{0})-wild partitions to Algebras_{F^{un}}^{σ}. In particular, λ_{F,n,c_{t},c_{w}} is the number of F^{un}-algebras of degree n, tame conductor c_{t}, and wild conductor c_{w}; this is the promised conceptual explanation of the integrality of λ_{F,n,c_{t},c_{w}}.

Explicit bijections For wild partitions to truly index geometric algebras, one would need to choose explicit σ-invariant bijections from W (p, e_{0}, e, s) to Fields_{F^{un}}(e, s) for each (e, s). We do not need explicit bijections for our purposes, but we describe the simple case F = ℚ_{p} and e = p to give a first indication of how the general case would look. Our description is taken from [2], which also describes the case e = p for general F. In our setting, the possible Swan conductors are Ore(p, 1, p) = {1, …, p}. Always the associated dimension is d(p, 1, p, s) = 1. If s < p then an explicit σ-equivariant bijection is

W (p, 1, p, s) = \overline{𝔽}_{p}^{×} → Fields_{F^{un}}(p, s),
a → ℚ_{p}^{un}[x]/(x^{p} + pãx^{s} + p).

For s = p, an explicit σ-equivariant bijection is

W (p, 1, p, p) = \overline{𝔽}_{p} → Fields_{F^{un}}(p, p),
a → ℚ_{p}^{un}[x]/(x^{p} + p + p^{2}ã).

In each case, ã ∈ ℤ_{p}^{un} is any lift of a.

Internal structure of geometric packets. Let L ∈ Algebras_{F^{un}}^{σ}. In many cases, the corresponding geometric packet {K_{1}, …, K_{g}} ⊂ Algebras_{F} consists of a single algebra of mass one. For example, suppose L has degree p and Swan conductor not divisible by p − 1. Then automatically K_{1}/F is not a Galois extension [2] and this forces |Aut(K_{1}/F)| = 1.

On the other hand, in many other cases g > 1. For example, let L be a product of m factors of F^{un}. Then the geometric packet of models for L consists of algebras F_{μ}, where μ is a partition of m. Here if μ = μ_{1} + ⋯ + μ_{h} then F_{μ} = F_{μ_{1}} × ⋯ × F_{μ_{h}}, where, as before, F_{f} denotes the degree f unramified extension of F. Then |Aut(F_{μ}/F)| = ∏_{k} k^{m_{k}}m_{k}!, where m_{k} is the number of times k appears in μ and Equation (61) becomes the class equation for the symmetric group S_{m}.

When a packet {K_{1}, …, K_{g}} contains a totally ramified field then all its elements are totally ramified fields. The database of local fields [12] contains many instances. For example, suppose K_{1} is a sextic field with automorphism group S_{3} and hence mass 1/6. Then its packet is {K_{1}, K_{2}, K_{3}} where K_{2} has mass 1/2 and K_{3} has mass 1/3. For i = 1, 2, 3, the corresponding Galois closures K_{i}^{g} have Galois group Gal(K_{i}^{g}/F) with S_{3} as inertia subgroup and the cyclic group C_{i} as corresponding quotient. For F = ℚ_{3}, the database presents five such packets.

The packets just discussed correspond to the irreducible partitions of Section 2. More generally suppose a packet {K_{1}, …, K_{g}} contains a field with residual degree f. Then the K_{i} which are fields all have residual degree f and their total mass is 1/f. These packets correspond to the isotypical partitions of Section 2.

11 Number fields

The sets Fields_{F,n,S}. Let F [x] = ℚ[x]/g_{0}(x) be a number field. Let S(F) be its set of places, indexing the set of completions of F. Thus S(ℚ) = {∞, 2, 3, 5, … } and S(F) maps surjectively to S(ℚ).

For S ⊆ S(F), let Fields_{F,n,S} be the set of isomorphism classes of degree n field extensions K/F ramified entirely within S. The ramification condition is then that for all v ∈ S(F)−S, and all w ∈ S(K) over S, the local extension K_{w}/F_{v} is unramified. In this context, we view ℂ/ℝ as ramified.

An extension K/F has a Galois closure K^{g} and hence a Galois group Gal(K^{g}/F); if K = F [x]/g(x), then K^{g} is by definition a splitting field of g(x). The largest that Gal(K^{g}/F) can be is the full symmetric group S_{n}, with n = [K : F]. For n ≥ 3, the second largest that Gal(K^{g}/F) can be is the alternating group A_{n}. We have a decomposition

Fields_{F,n,S} = Fields_{F,n,S}^{sym} ∐ Fields_{F,n,S}^{alt} ∐ Fields_{F,n,S}^{small}  (64)

and also write Fields_{F,n,S}^{big} to indicate the union of the first two parts. We write also Fields_{F,S}^{s} = ∐_{n} Fields_{F,n,S}^{s} for any superscript s. In practice, it is easy to decide whether a given field F [x]/g(x) has big or small Galois group. One quick way is to factor g(x) in sufficiently many completions F_{v} and use information from the degrees of the factor fields K_{w}; for most v, this reduces to a calculation in the residue field of F_{v}. For n ≥ 8, a group-theoretical result of Jordan [9] suffices: the Galois group is big if and only if

the degree of K_{w}/F_{v} is a prime in (n/2, n − 2)  (65)

for some K_{w}/F_{v}. Many other criteria can be brought to bear as well. The computations are guided by the principle that the factor partitions for v not ramified in K are equidistributed in the set of partitions of n according to the measure induced from the Haar measure on Gal(K^{g}/F).

We are interested in the case of S finite. Then a classical fact is that the sets Fields_{F,n,S} are all finite. Analogously to the local situation, it is natural to define the mass of a field K to be 1/|Aut(K/F)|. From (64) we have ϕ_{F,n,S} = ϕ_{F,n,S}^{big} + ϕ_{F,n,S}^{small}. Our main concern is ϕ_{F,n,S}^{big}, which is just the cardinality of Fields_{F,n,S}^{big} when n ≥ 4.

For S all of S(F), a principle in number theory is that the group S_{n} is very common, in many rigorous senses. One might at first expect that the sets Fields_{F,n,S} would behave like smaller versions of the set Fields_{F,n,S(F)}, so that most fields in Fields_{F,n,S} would be in Fields_{F,n,S}^{sym}. This section argues that the evidence points in the opposite direction, at least when one fixes S and considers all n simultaneously.

Ease of constructing fields in Fields_{F,S}^{small}. One has Fields_{ℚ,S} = {ℚ} for S = {} or S = {∞}. Otherwise, Fields_{ℚ,S} is infinite, as it at least contains the real cyclotomic fields ℚ[x]/Φ_{p^{k}}^{+}(x) for all p in S and all positive k. Since the Galois group of ℚ[x]/Φ_{p^{k}}^{+}(x) is abelian, these fields are in Fields_{ℚ,S}^{small} whenever their degree is ≥ 4.

There is an elaborate theory for describing the part of Fields_{F,S}^{small} consisting of fields K such that Gal(K^{g}/ℚ) is solvable [14, 18]. This theory says that as soon as S is large enough, Fields_{F,S}^{small} is very large indeed. For example, let L be a maximal pro-2-extension of ℚ ramified only within S = {∞, 2}. Then Markshaitis’ theorem [14, Example 11.18] says Gal(L/ℚ) is the free pro-2 product of ℤ/2 and ℤ_{2}. Accordingly ϕ_{ℚ,2^{k},{∞,2}}^{small} grows exponentially with n = 2^{k}.

There are also general techniques for constructing non-solvable fields in Fields_{F,S}^{small}. For example using modular forms gives fields with Galois groups with P SL_{2}(𝔽_{ℓ^{f}}) as a simple subquotient. Already this technique shows that the Galois group corresponding to any Fields_{ℚ,{∞,p,ℓ}} has infinitely many simple subquotients different from A_{n}. The ABCconstruction of [20] shows that one can likewise expect infinitely many simple subquotients of the form P Sp_{2k}(𝔽_{ℓ}) involved in Fields_{ℚ,S} for S large enough, e.g. S = {∞, 2, 3}.

Difficulty of constructing fields in Fields_{F,S}^{big}. All known constructional techniques for fields with Galois group all of A_{n} or S_{n} have only modest control over ramifying primes. The most well-known technique, and one of the best, is uses trinomials. For example, take F = ℚ and for t ∈ ℚ − {0, 1} consider the polynomial

g_{n,t}(x) = x^{n} − ntx + (n − 1)t.  (66)

Its discriminant is

D_{n,t} = (−1)^{(n−1)(n−2)/2}n^{n}(n − 1)^{n−1}t^{n−1}(t − 1).  (67)

Its Galois group is generically S_{n} or A_{n} according to whether or not D_{n,t} is a square. If one chooses t such that the denominator of t and the numerator of t and t − 1 are only divisible by primes dividing n(n − 1), then only these primes can ramify in K_{n,t} = ℚ[x]/g_{n,t}(x). The problem here is that only finitely many n keep ramification within any given S. Even in the more general context of arbitrary trinomials described in [20, Section 10], with two relatively prime parameters n > m, there are only finitely many (n, m) such that all primes dividing nm(n − m) are within S. The recent technique of Chebyshev covers [21] gives larger degree fields, but still to get fields ramified within a given S, one needs an appropriate “numerical accident” such as 2^{3} + 1 = 3^{2} for S = {∞, 2, 3}.

The finiteness conjecture. Based on the considerations just presented, we make the following conjecture.

Conjecture 11.1. Let F be a number field and let S be a finite set of places of F. Then the set Fields_{F,S}^{big} is finite.

In other words, while ϕ_{F,S}^{small} is usually infinite, we expect ϕ_{F,S}^{big} to always be finite.

Heuristic support for the finiteness conjecture. Bhargava [4] has a heuristic formula for the “expected number” of A_{n} and S_{n} fields in a given degree n with a given discriminant d. The asymptotic behavior of Bhargava’s heuristic as |d| → ∞ agrees with the previously known Davenport-Heilbronn theorem in n = 3. In fundamental work, Bhargava has proven the analogous theorem for n = 4 [3] and has announced it for n = 5, giving one moderate confidence in the heuristic formula for general n.

Applying Bhargava’s heuristic to our situation gives the following “expected number”

ϕ_{F,n,S}^{big} ≈ \frac{1}{2} ∏_{v∈S} λ_{F_{v},n},  (68)

where here we require that all Archimedean places are in S. Both λ_{ℝ,n} and λ_{ℂ,n} decrease superexponentially with n, according to (42). For each ultrametric place v of F, the sequence λ_{F_{v},n} increases only exponentially, with growth factor Q_{v}^{1/(p_{v}−1)}. All together, ∏_{v∈S} λ_{F_{v},n} decreases superexponentially with n. This is much more than the mere convergence of ∑_{n} ∏_{v∈S} λ_{F_{v},n} which would be enough to heuristically support Conjecture 11.1. Note that one has a heuristic product formula (68) only when one appropriately separates by Galois groups. For example, Bhargava’s results show that S_{4} and D_{4} need to be treated separately.

We understand the factor 1/2 in (68) in two different ways, depending on whether n = 2 or n ≥ 3. The case n = 2 is best first explained in the simplified setting F = ℚ and {∞, 2} ⊆ S. Then λ_{ℚ_{v},2} is 1, 4, or 2 according to whether v is ∞, 2, or otherwise. The set Fields_{ℚ,2,S} has exactly 2^{|S|} − 1 elements, each with mass 1/2. So its total mass is 2^{|S|−1} − 2^{−1} while (68) is 2^{|S|−1}. The agreement would be perfect if we worked instead with Algebras_{ℚ,2,S} to account for the trivial alternating group A_{2}. For general F, the 1/2 in the case n = 2 likewise comes from the fact that fields in Fields_{F,2,S} have mass 1/2 rather than the usual 1. For the cases n ≥ 3, one uses the local signs (2, d_{v})HW (K_{v}) associated to K/F ∈ Algebras_{F,n,S} and a place v ∈ S. While these signs are all 1 in the case n = 2, in general they can be 1 or −1.

For n ≥ 3, the 1/2 in the mass formula corresponds to the fact that the product of the local signs is 1 for any K in Fields_{F,n,S}. For more explicit information on these signs in the case F = ℚ, see [12, Section 3.3].

Comparison with computational results over ℚ. Figure 6 summarizes known facts about the case (F, S) = (ℚ, {∞, 2, 3}). For n = 1, 2, 3, 4, 5, 6, and 7, Jones and Roberts [10], [11] evaluated ϕ_{ℚ,\{2,3,∞\}}^{big} to 1, 3.5, 8.\overline{3}, 22, 5, 54, and 10. Roberts [20] found more fields in degrees n = 8, 9 and also the S_{32} field ℚ[x]/(x^{32} + 2^{16}3^{5}x^{5} + 2^{13}3^{9}) with discriminant 2^{191}3^{112}. Jones is finding more fields in degrees 8 and 9 by an ongoing computer search. Malle and Roberts [16] found 300 more fields in degree 9 ≤ n ≤ 33 and discussed the issue of finiteness of ϕ_{ℚ,S}^{big} noncommittally as an open question. Roberts [21] found 43 more fields in degrees 12 ≤ n ≤ 64.

Figure 6: Evaluations (black) and lower bounds (gray) for ϕ_{ℚ,n,{∞,2,3}}^{big}, compared with \frac{1}{2}λ_{ℝ,n}λ_{2,n}λ_{3,n}, with logarithmic vertical scale.

Low discriminant phenomena and exceptional fields. Figure 6 also compares the above computational results with the more theoretical quantity \frac{1}{2}λ_{ℝ,n}λ_{2,n}λ_{3,n}. Although we are confident in Conjecture 11.1 on a qualitative level, the situation remains enigmatic on a quantitative level.

We interpret the poor agreement in degrees ≤ 7 as reflecting the fact that Bhargava’s heuristic does not take into account low discriminant phenomena. Experimentally, these low discriminant phenomena always seem to give fewer fields, with the case of cubics quantitatively explained by Roberts [19] using a negative secondary term. Our guess is that the very poor agreement in medium degrees is due to two factors, the same low discriminant phenomena and the incompleteness of the current list of fields.

On the other hand, the poor agreement in degree 64 is in the other direction. We interpret this disagreement as an indication that the constructional method of [21] is very special in nature. Define a field in Fields_{F,n,S}^{big} to be exceptional if ϕ_{F,n′,S}^{big} < 1 for all n′ ≥ n. The starting point N(F, S) of the exceptional range is not as artificial as may first seem, because the decay of ϕ_{F,n,S}^{big} is rapid once it begins.

For F = ℚ and S = {∞, 2, 3}, {∞, 3, 5}, and {∞, 2, 5} the exceptional range starts at N(ℚ, S) = 62, 38, and 49 respectively. The field constructed in [21] of degree 100, Galois group A_{100}, and discriminant of the form 3^{a}5^{b} is well into the exceptional range. Similarly, the five fields constructed there of degrees 2666 through 15875 and discriminant of the form ±2^{a}5^{b} are exceptional if, as strongly expected, their Galois groups are the full symmetric group on the degree.

Comparison with computational results over quadratic fields. In general, let S be a set of rational places containing ∞. Let F be any degree n_{0} number field. For v a place of ℚ, let λ_{F_{v},n} = ∏ λ_{F_{w},n}, the product being over places w mapping to v. Then λ_{F_{∞},n}^{1/n} ∼ (e/n)^{n_{0}}, independently of the splitting behavior of ∞ in F. Similarly, λ_{F_{p},n}^{1/n} ∼ p^{n_{0}/(p−1)} independently of the splitting behavior of p. However, looking at the subexponential factors suggests that λ_{F_{v},n} is at its lowest if F_{v} is a field and increases substantially as F_{v} tends towards the split algebra ℚ_{v}^{n_{0}}.

To illustrate this in practice, let F be the field F_{d} = ℚ(\sqrt{d}), where d varies over the set {−6, −3, −2, −1, 2, 3, 6}. Let S_{d} ⊂ S(F_{d}) be the set of places mapping to ∞, 2, or 3 in S(ℚ). So |S_{d}| = 3 for d ∈ {−6, −3, −1} as none of ∞, 2, or 3 is split in F_{d}. In contrast, |S_{d}| = 4 if d ∈ {−2, 2, 3, 6} as 3 is split in F_{−2} and ∞ is split in the remaining cases.

Table 3: Evaluations and lower bounds for ϕ_{ℚ(\sqrt{d}),n,\{∞,2,3\}_{d}}^{big} in italics
under d compared with \frac{1}{2}λ_{ℚ(\sqrt{d})_{∞},n}λ_{ℚ(\sqrt{d})_{2},n}λ_{ℚ(\sqrt{d})_{3},n} in plain type under
∏.
No v split  3 split  ∞ split
n  ℂ ℝ · ℝ 4 2 · 2 9 3 · 3 −6 −3 −1 ∏ −2 ∏  2  3  6 ∏
2 0.500 1.000 8 16 2 4 3 .5 3 .5 3 .5 4 7 .5 8 7 .5 7 .5 7 .5 8
3 0.167 0.444 9 25 27 81 14 .\overline{3} 14 .\overline{3} 14 .\overline{3} 20 46 .\overline{3} 61 40 .\overline{3} 40 .\overline{3} 40 .\overline{3} 54
4 0.042 0.174 272 1296 29 121 87 87 87 164 343 686 385 385 385 685
5 0.008 0.047 280 1600 55 361  17 21 64  421 ≥ 87  361

Table 3 compares the two sides of (68) in degrees 2 ≤ n ≤ 5. The first pair of columns gives λ_{ℂ,n} < λ_{ℝ,n}^{2} to three decimal places. The next two pairs of columns likewise give λ_{4,n} < λ_{2,n}^{2} and λ_{9,n} < λ_{3,n}^{2}. The column λ_{2,n}^{2} is given for the sake of uniformity, but is not needed as 2 does not split in any of the F_{d}. Next follow masses ϕ_{ℚ(\sqrt{d}),n,\{∞,2,3\}_{d}}^{big} in italics, with the corresponding product \frac{1}{2}λ_{ℚ(\sqrt{d})_{∞},n}λ_{ℚ(\sqrt{d})_{2},n}λ_{ℚ(\sqrt{d})_{3},n} rounded to the nearest integer in regular type to its right. The italicized masses are computed from [10] for n = 2, 3 fields and from [5] and associated ongoing searches for n = 4, 5. For each n, the references list degree 2n extensions of ℚ with a corresponding degree 2n permutation group G. We extracted those with subfields isomorphic to F_{d}. Each such field K gives either a single extension of F_{d} or two conjugate extensions, according to whether the Galois closure K^{g} of K over F_{d} is Galois or not over ℚ. The Galois groups G giving one and two fields are as follows:

2n One field  Two conjugate fields
4 C_{4}, V  D_{4}
6 A_{3}C_{2}^{∗}, S_{3}^{t}C_{2}^{∗}, S_{3}C_{2} T 5^{∗}, T 9, T 10, T 13
8 A_{4}C_{2}^{∗}, S_{4}^{t}C_{2}^{∗}, S_{4}C_{2} T 33^{∗}, T 34, T 41, T 42^{∗}, T 45, T 46, T 47
10 A_{5}C_{2}^{∗}, S_{5}^{t}C_{2}^{∗}, S_{5}C_{2} T 40^{∗}, T 41, T 42, T 43

Here the last group on each list is always S_{n}^{2}.C_{2}. The starred groups are the ones giving rise to Gal(K^{g}/ℚ(\sqrt{d})) ≅ A_{n}. This distinction plays a role in the construction of Table 3 only for n = 3, as in this case A_{3} fields are counted with mass 1/3; if we were counting the total number of fields then the n = 3 entries left to right would be 17 , 23 , 17 , 49 , 41 , 41 , and 43 .

Table 3 reflects again how (68) does not take into account low discriminant phenomena and consequently \frac{1}{2} ∏_{v∈S} λ_{ℚ(\sqrt{d})_{v},n} is an overestimate for the total mass ϕ_{F_{d},n,S_{d}}^{big}. However, Table 3 also illustrates a uniformity in the overestimation, as in degrees 2, 3, 4, and 5 the factor is approximately 1, 1.3, 2, and 3.5, independent of d. Thus the principle that splitting primes in the base field increase the number of fields is clearly visible at the level of actual fields.

12 Contrast with positive characteristic

It is standard in number theory to talk about global fields, meaning either number fields as in Section 11 or function fields in one variable over a finite field. By the latter, one means fields F which can be presented as finite extensions of some rational function field 𝔽_{q}(t). One can talk uniformly about completions of these global fields, getting local fields. In characteristic zero, the local fields are exactly the ones considered in Sections 7-10. In positive characteristic p, all local fields are isomorphic to the Laurent series field 𝔽_{q}((u)), for q some power of p. Often in number theory, the characteristic zero and the positive characteristic situations are quite similar; this principle is well illustrated by our comments on the Serre mass formula below. However with regards to both our specialization (y, z) = (1, 1) and our finiteness conjecture, the case of positive characteristic presents a sharp contrast.

Allowing e_{0} = ∞. To accommodate positive characteristic, Sections 2 and 3 can be extended by allowing e_{0} = ∞ also. Ore numbers in this situation are simpler. Namely, as before, Ore(p, ∞, e) is {0} in the tame case when e is not a multiple of p. However in the complementary wild case, the Ore table consists of a single block with infinitely many rows. Thus Ore(p, ∞, e) is always the complete set of positive integers which are not multiples of p; this is a radical difference because now the number of Ore numbers is infinite. The dimension associated to an Ore number s is simply d(p, ∞, e, s) = ⌈s/p⌉. Since all Ore numbers s are non-maximal, W (p, ∞, e, s) always consists of the vectors in \overline{𝔽}_{p}^{d(p,∞,e,s)} with first component nonzero. The function Λ_{p,∞,f_{0}}(x, y, z) is defined as before. Writing s = s_{0}p + s_{1}, with 1 ≤ s_{1} ≤ p − 1, it takes the explicit form

Λ_{p,∞,f_{0}}(x, y, z) = ∏_{s_{0}=0}^{∞} ∏_{s_{1}=1}^{p−1} \frac{1 − q^{s_{0}}x^{e}y^{e−1}z^{ps_{0}+s_{1}}}{1 − q^{s_{0}+1}x^{e}y^{e−1}z^{ps_{0}+s_{1}}} ,  (69)

with q = p^{f_{0}}, as always.

A contrast between the two specializations. The two situations in Section 4 now present a sharp contrast. The Serre mass formula continues to hold: Λ_{p,∞,f_{0}}(x, 1, 1/q) = Λ(x), with essentially the same proof. Our formula Λ_{p,∞,f_{0}}(x, 1, 1) = Λ_{Q}(x) becomes completely degenerate as Q = p^{e_{0}f_{0}} needs to be regarded as ∞. The coefficient of x^{n} on both sides is λ_{n} if n < p and ∞ otherwise. There is no reasonable analog of Section 5 or 6.

Similar local behavior. Section 7-10 go through in the positive characteristic setting so that in particular the field-theoretic quantity Λ_{F} (x, y, z) agrees with the directly defined Λ_{p,∞,f_{0}}(x, y, z) of (69). Ramification can be understood from a different perspective in the characteristic p setting. Namely if K/F = 𝔽_{q}((v))/𝔽_{q}((u)), then one can write u = ∑_{i=1}^{∞} b_{i}v^{i}. The smallest i such that b_{i} ≠ 0 is just the degree n = e. The smallest i not divisible by p such that b_{i} ≠ 0 is 1 + c, where c = c_{t} + c_{w} = (n − 1) + c_{w}; this i is just the first exponent such that the corresponding derivative ib_{i}v^{i−1} is non-zero.

Ease of constructing of fields in Fields_{F,n,S}^{big}. The statement of Conjecture 11.1 makes sense with “number field” replaced by “function field” but it is false in general. As a general source of field extensions of F = 𝔽_{p}(t), take

g(x) = a(x)^{p} + x + b(t)  (70)

with a(x) and b(t) running over polynomials with coefficients in 𝔽_{p}. One has always g′(x) = 1 and so the polynomial discriminant (43) is (−1)^{n(n−1)/2} with n the degree of g(x). Thus F [x]/g(x) defines a separable algebra which can only be ramified at the the “infinite” place F_{∞} = 𝔽_{p}((1/t)) of 𝔽_{p}(t). If the algebra F_{∞}[x]/g(x) has conductor c then F [x]/g(x) is the function field of a curve of genus 1 − n + c/2 by the Riemann-Hurwitz formula.

As a simple counterexample to the analog of Conjecture 11.1, consider the very special case g_{k}(x) = x^{2} + x + t^{2k+1} for k a non-negative integer. Changing variables via x = 1/(ys^{k}) and t = 1/s the equation becomes y^{2} + s^{k+1}y + s. This is an Eisenstein polynomial over the ring 𝔽_{2}[[s]] with conductor c = c_{t}+c_{w} = (1)+(2k+1) = 2k+2. So F [x]/(x^{2}+x+t^{2k+1}) is the function field of a genus k curve. Thus the F [x]/g_{k}(x) form an infinite collection of fields in Fields_{F,2,{∞}}^{big}. For general p and a(x) of degree k = 1, the polynomial g(x) = a(x)^{p} + x + b(t) likewise defines an Artin-Schreier extension of 𝔽_{p}(t). For k > 1 we expect that “almost all” specializations of the family (70) to extensions of 𝔽_{p}(t) have Galois group all of A_{n} or S_{n}.

Another simple specialization of (70) is g_{p,k}(x) = x^{kp} −x+t for p a prime and k a positive integer. Changing variables via x = 1/y and t = 1/s, the equation becomes y^{kp} − sy^{kp−1} + s. This is an Eisenstein polynomial with conductor c = c_{t} + c_{w} = (kp − 1) + (kp − 1) = 2kp − 2. Thus 𝔽_{p}(t)[x]/g_{p,k}(x) is the function field of a genus zero curve, a fact which is also clear by the single global observation that F [x]/g_{p,k}(x) = 𝔽_{p}(x), as t = −x^{kp} − x. If k = p^{j} then one has g_{p,p^{j}}(x) = h_{j}(x) + t with h_{j}(x) the Artin-Schreier polynomial h_{1}(x) = x^{p} − x composed with itself j times. Thus the Galois group of g_{p,p^{j}}(x) is solvable. In the complementary case, computations with Frobenius elements using Jordan’s criterion (65) suggest that the Galois group of g_{p,k}(x) is all of A_{pk} or S_{pk} except in the cases g_{3,4}(x) = x^{12} − x + t and g_{2,12}(x) = x^{24} − x + t. Here the Galois groups are known to be the Mathieu group M_{11} in its degree 12 representation and the Mathieu group M_{24}, respectively [1, Theorems 6.6 and 6.3]. So the evidence is strong that for each p, the field 𝔽_{p}(t)[x]/g_{p,k}(x) is in Fields_{𝔽_{p}(t),pk,{∞}}^{big} for infinitely many k. Abhyankar has studied many similar genus zero families; typically the focus is extracting rare examples with small Galois groups from families with generic Galois group A_{n} or S_{n}.

The same heuristic which supports Conjecture 11.1 in characteristic zero gives two reasons why the corresponding statement fails in positive characteristic. Again one can interpret \frac{1}{2} ∏_{v∈S} λ_{F_{v},n} as the expected total mass of fields in Fields_{F,n,S}^{big}. But in the function field case if n ≥ p then each of the factors λ_{F_{v},n} is itself infinite; this is the phenomenon behind the existence of the family g_{k}(x) above. Even if one bounds ramification somehow so that each λ_{F_{v},n} is replaced by a finite number λ_{F_{v},n}^{∗}, the numbers \frac{1}{2} ∏_{v∈S} λ_{F_{v},n}^{∗} can still increase due to the lack of Archimedean places. This is the phenomenon behind the existence of the family of polynomials g_{p,k}(x).

References

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2000 Mathematics Subject Classification: Primary 11S15; Secondary 11P72, 11R21. Keywords: wild, partition, p-adic, ramified, mass.

(Concerned with sequences \underline{A000041}, \underline{A000085}, \underline{A010054},\underline{A033687}, \underline{A131139}, and \underline{A131140}.) Received March 28 2007; revised version received June 18 2007. Published in Journal of Integer Sequences, June 18 2007. Return to Journal of Integer Sequences home page.