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\begin{center}
\vskip 1cm{\LARGE\bf 
Wild Partitions and Number Theory}
 \\ 

\vskip 0.5 cm
\large
David P. Roberts\\
Division of Science and Mathematics\\
University of Minnesota, Morris\\
Morris, MN, 56267 \\
USA\\
\href{mailto:roberts@morris.umn.edu}{\tt roberts@morris.umn.edu}\\
\end{center}

\vskip .2in

\begin{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 $\lambda_{Q,n}$ counting the number of certain  
wild
partitions of $n$.  We give an explicit formula for the 
  corresponding generating function $\Lambda_Q(x) = \sum \lambda_{Q,n} x^n$ and use
  it to show that  $\lambda^{1/n}_{Q,n}$ 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.  
\end{abstract}


\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{conjecture}{Conjecture}[section]
\newtheorem{corollary}{Corollary}[section]
\newtheorem{lemma}{Lemma}[section] 
\newtheorem{definition}{Definition}[section]


\section{Introduction}
\label{Intro}


          The sequence \seqnum{A000041} of integers $\lambda_n$ giving the number of partitions of $n$ is important throughout mathematics.   Its generating function is
 \begin{equation}
 \label{part}
\Lambda(x) =  \sum_{n=0}^\infty \lambda_n x^n = \prod_{e=1}^\infty \frac{1}{1-x^e} = 
1 + x + 2 x^2 + 3 x^3 + 5 x^4 + 
 7 x^5 + \cdots .
 \end{equation}
 In this paper, we consider for any prime power $Q = p^{n_0}$, an  
 analogous integer sequence $\lambda_{Q,n}$
 arising in a fundamental way in the number theory of $p$-adic fields.   
 We evaluate the associated generating functions $\Lambda_Q(x) = \sum \lambda_{Q,n} x^n$,
  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 \cite{Sl} as \seqnum{A131139} and \seqnum{A131140}, give a first sense of the functions 
  $\Lambda_{Q}(x)$ and the corresponding sequences $\lambda_{Q,n}$:
  \begin{eqnarray*}
 \Lambda_2(x) & = & \frac{1}{1-x} \cdot \frac{1-x^2}{\left(1-2 x^2\right)^2} \cdot \frac{1}{1-x^3} \cdot 
 \frac{  \left(1-x^4\right) \left(1-4 x^4\right)^2 }{\left(1-8 x^4\right)^4} \cdot \frac{1}{1-x^5} \cdot \frac{1-x^6}{\left(1-8 x^6\right)^2} \cdot \;  \cdots \\
              & = & 1 + x + 4 x^2 + 5 x^3 + 36 x^4 + 40 x^5 + 145 x^6 + 180 x^7 + 1572 x^8 + 1712 x^9 + \cdots ,\\
              &\\
 \Lambda_3(x) & = & \frac{1}{1-x} \cdot \frac{1}{1-x^2} \cdot \frac{\left(1-x^3\right)^2}{\left(1-3 x^3\right)^3} \cdot \frac{1}{1-x^4} \cdot \frac{1}{1-x^5} \cdot  \frac{\left(1-x^6\right)^2}{\left(1-9 x^6\right)^3} \cdot \;  \cdots  \\
 & = & 1 + x + 2 x^2 + 9 x^3 + 11 x^4 + 19 x^5 + 83 x^6 + 99 x^7 + 172 x^8 + 1100 x^9 + \cdots .
 \end{eqnarray*}
 In general, $\Lambda_Q(x)$, like its model $\Lambda(x)$, 
  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
 {\em tame} ramification and the latter reflect {\em 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 \cite{JR} and 
would
considerably facilitate the $p$-adic analysis of number fields.  

       Sections~\ref{Wild}-\ref{One} are our combinatoric sections.    
       For a factorization $n_0 = e_0f_0$, we define 
{\em $(p,e_0,f_0)$-wild partitions} and a corresponding complicated three-variable
generating function $\Lambda_{p,e_0,f_0}(x,y,z)$.   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
 \begin{align}
  \label{serrespec}
 \Lambda_{p,e_0,f_0}(x,1,p^{-f_0}) & = \Lambda(x).
 \end{align}
 As we'll indicate, this identity is related to the Serre mass formula \cite{Se}.
 Our main interest is in the new specialization $(y,z) = (1,1)$. 
 We define $\Lambda_Q(x)$ by an explicit formula and find
 \begin{align}
  \label{myspec}
  \Lambda_{p,e_0,f_0}(x,1,1) & = \Lambda_Q(x), 
    \end{align}
 independently of the factorization $n_0 = e_0 f_0$.   Our explicit formula allows us to consider
 arbitrary real powers $Q = p^\nu \geq 1$ so that $Q$ no longer determines $p$ and 
 we accordingly write $\Lambda_{p,Q}(x)$.   One has $\Lambda_{p,1}(x) = \Lambda(x)$,
 independently of $p$.  Thus another point of view is that for each prime $p$ we have 
 a $Q$-analog of $\Lambda(x)$.  
   
      Sections~\ref{Core} and \ref{Asymptotics} consider analytic number theory associated to
 $\Lambda_{p,Q}(x)$.
 We express $\Lambda_{p,Q}(x)$ directly in terms of $\Lambda(x)$
 and observe that as a consequence   
 \begin{equation}
 \label{rootlimit}
 \lim_{n \rightarrow \infty} \lambda_{p,Q,n}^{1/n} = Q^{1/(p-1)}.
 \end{equation}
 Thus $\lambda_{p,Q,n}$ grows exponentially with growth factor $Q^{1/(p-1)}$.
 Equation~\eqref{rootlimit} contrasts with the famous 
 Hardy-Ramanujan statement \cite{HR} of subexponential growth, 
 $\lambda_{n}  \sim e^{\pi \sqrt{2 n/3}}/(4 n \sqrt{3})$.  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 $\lambda_n/\lambda_{n-1}$ tends to $1$ while    $\lambda_{p,Q,n}/\lambda_{p,Q,n-1}$ has oscillatory behavior which
becomes more pronounced as $Q$ increases.  
 We conjecture an asymptotic of the form
 \begin{equation}
 \label{asymphope1}
 \lambda_{p,Q,n} \sim c_{p,Q}(n) C_p(Q) n^{B_p(Q)} e^{A_p(Q) \sqrt{n}} Q^{n/(p-1)},
 \end{equation}
 with an explicit factor $c_{p,Q}(n)$ capturing the oscillatory behavior of  
 $\lambda_{p,Q,n}/\lambda_{p,Q,n-1}$.
 
 
 

      
    
        Sections~\ref{RC}-\ref{adic2} 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~\ref{RC} sets up the general situation and illustrates 
  it for the fields $\R$ and $\C$, getting simple functions $\Lambda_\R(x)= e^x$ and
  $\Lambda_\C(x) = e^{x+x^2/2}$ which serve as analogs of our $\Lambda_Q(x)$.    
  For Sections~\ref{Eisenstein}-\ref{adic2},
  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~\ref{adic1}
  explains how the coefficient $\lambda_{n,c_t,c_w}$ 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~\ref{adic2} works over the maximal
  unramified extension $F^{\rm un}$ of $F$.  It explains how 
  $\lambda_{n,c_t,c_w}$ also counts extension algebras of $F^{\rm un}$ with
  the corresponding invariants.   The perspective of Section~\ref{adic1} is
  more directly connected with the literature, while the perspective of
  Section~\ref{adic2} explains why the coefficients of $\Lambda_{p,e_0,f_0}(x,y,z)$
  are integers.     Summing over the possible $c_t$ and $c_w$, one gets that
  the total mass $\lambda_{F,n}$ of degree $n$ extension algebras of $F$ is 
  $\lambda_{Q,n}$, where $Q = p^{n_0}$.  
  

  

  
       Section~\ref{NumberFields} shifts to global algebraic number theory, working over
   an arbitrary number field $F$.  It addresses
    a   question raised in \cite{MR} on the size of sets $\mbox{Fields}^{\rm big}_{F,n,S}$ 
    of relative degree $n$ number fields $K/F$.  To be in $\mbox{Fields}^{\rm big}_{F,n,S}$, 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 \cite{Bh} yields 
    \begin{equation}
    \label{Bhargintro}
    \frac{1}{2} \prod_{v \in S} \lambda_{F_v,n}
    \end{equation}
    as a first guess (after slight modifications in degrees $\leq 3$) 
    for the size of $\mbox{Fields}^{\rm big}_{F,n,S}$. 
    The Archimedean factors $\lambda_{\R,n}$, $\lambda_{\C,n}$ decay superexponentially
    and we have proved that the remaining $\lambda_{F_v,n}$ grow only exponentially.  
    Thus \eqref{Bhargintro} leads to the 
     prediction that for fixed $(F,S)$, the set $\mbox{Fields}^{\rm big}_{F,n,S}$ is
    empty for sufficiently large $n$.     In other words
     $\mbox{Fields}^{\rm big}_{F,S} = \coprod_n \mbox{Fields}^{\rm big}_{F,n,S}$ 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~\ref{Contrast} 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 $\infty$ and this forces all the $\lambda_{F_v,n}$ appearing
 in \eqref{Bhargintro} to be infinite for $n \geq p$.    Another source is that there are no
 Archimedean places of $F$, and thus no superexponentially decaying factors
 in \eqref{Bhargintro}.  

       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~\ref{RC}-\ref{adic2} are then 
 only the completions of $\Q$, i.e.\ $\R$ and the $\Q_p$.  
 The ground fields $F$ of Section~\ref{NumberFields} are then
 limited to simply $\Q$ itself.    However by explicit examples with $n_0 = 2$ in 
 Sections~\ref{Wild}, \ref{Three}, \ref{Asymptotics}, and \ref{NumberFields},
 we try to assist readers in appreciating the case of general $(e_0,f_0)$.  
 Sections~\ref{RC}-\ref{NumberFields} 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~\ref{Core} and \ref{Asymptotics} which interpolate $n_0 = e_0 f_0 \in \Z_{\geq 1}$ 
 with general reals $\nu \geq 0$.  The naturality of the general setting is further underscored by
  Section~\ref{Contrast}, which is based on
 the limiting case $e_0 = \infty$.  
 
      

 
 
 
 
 
 
   
     
\section{Wild partitions} 
\label{Wild}

\paragraph{Basic notation.}  An ordinary partition is an element of the free abelian monoid generated by
the set  of allowed parts $P = \{1,2,3,\dots\}$, for example
\begin{equation}
\label{exampleordinary}
\mu_{\rm ordinary} = 9 + 7 + 3 + 3 + 2 + 2 + 2 +  1.
\end{equation}
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 $\sigma = \sigma_p^{f_0}$ 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 $n_0 = e_0f_0$.
Our notations $e_0$, $f_0$, $n_0$ and $q$ come from standard notations in the number-theoretic
situation of Sections~\ref{Eisenstein}-\ref{adic2}
 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 $\F_p = \Z/p \Z$.   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 
 $\sigma_{p}$ the Frobenius element $k \mapsto k^p$ in $\Gal(\overline{\F}_p/\F_p)$.  
 Similarly, we denote by $\sigma_{p^u}$ the element $\sigma_p^{u}$; it is a 
 topological generator of $\Gal(\overline{\F}_p/\F_{p^u})$.  Most important for us is
 the operator $\sigma_q$, which we abbreviate by simply $\sigma$.  
    


    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 {\em wildness} of $e$, $p^{w}$ as the {\em wild part} of $e$, and 
 $t$ as the {\em tame part} of $e$.  As another standing convention, we abbreviate $e_0 e$ by
 $E$.  

 \paragraph{Ore numbers and their associated dimensions and spaces.}     
 An important notion in number theory is the set of
{\em Ore numbers}
\begin{equation}
\Ore(p,e_0,e) \subseteq \{0,1,\dots,w E - 1,w E\}.
\end{equation}
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{equation}
\label{orefirst}
\begin{array}{ccccccccc}
. &    &     &   &     &     &    &    &     \\
\hline
. & 8 & 7 & .  & 5 & 4 & .  & 2 & 1  \\
. & 17 & 16 & . & 14 & 13 & . & 11 & 10  \\
\hline
. & 26 & 25 & 24 & 23 & 22 & 21 & 20 & 19  \\
36 & 35 & 34 & 33 & 32 & 31 & 30 & 29 & 28  \\
\end{array}
\end{equation}
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 $1 \leq j \leq w-1$, the $j^{\rm th}$ block consists
of the integers in the interval $[(j-1) E+1,j E]$ which are not multiples of $p^j$.  The $w^{\rm th}$
block consists of these entries together with $wE$.   Considering the table as a whole,
we refer to all the entries as {\em non-maximal}, except for $wE$ which is {\em maximal}.  
Our array format, including the right-to-left order, 
is intended to facilitate the discussion in 
Section~\ref{Eisenstein}, where the number-theoretic origin 
of $\Ore(p,e_0,e)$ is explained.  


       An important quantity in our situation is the {\em dimension} $d(p,e_0,e,s)$ associated to
 an Ore number $s \in \Ore(p,e_0,e)$.   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 \eqref{orefirst}.    The way dimensions arise in number theory
is explained in Section~\ref{adic1}.  

       An Ore number $s \in \Ore(p,e_0,e)$ 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 $s = w E$, 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 $\sigma = \sigma_p^{f_0}$ acts 
 coordinate-wise on each $W(p,e_0,e,s)$, as indeed $\sigma_p$ itself acts. 
 The number of fixed points of $\sigma$ 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 $s = w E$.   The explicit $\sigma$-sets
 $W(p,e_0,e,s)$ just introduced are isomorphic to less explicit 
 $\sigma$-sets arising naturally in number theory,
 as explained  in Section~\ref{adic2}.   We use $s$ as our variable 
 running over Ore numbers, because Ore numbers are also called Swan conductors.  

     
    


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

\begin{definition}
\label{combdef}
 A $(p,e_0,f_0)$-wild partition is an element of the free abelian 
monoid on the set
\begin{equation}
P(p,e_0) = \coprod_{e \in \Z_{\geq 0}} \coprod_{s \in {\rm Ore}(p,e_0,e)} W(p,e_0,e,s)
\end{equation}
which is fixed by 
 $\sigma = \sigma_p^{f_0}$.  
\end{definition}
\noindent 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 
$e_{s;\omega}$,  with $s \in \Ore(p,e_0,e)$ and 
$\omega \in W(p,e_0,e,s)$.   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 
\begin{equation}
\label{examplewild}
\mu_{\rm wild} = 9_{20; 1,1,0,2,2,0,1} + 7 + 3_{1;i} + 3_{1;-i} + 2 + 2 + 2 + 1
\end{equation}
is a wild partition for $(p,e_0,f_0) = (3,2,1)$.   To check that $\mu_{\rm wild}$ is indeed
 formed according to our rules, note that 
 $d(3,2,9,20) = |\{0,3,6,9,12,15,18\}| = 7$ from
\eqref{orefirst}, and 
so it is proper that first subscripted $\omega$ has length $7$.   Also 
the first coordinate of this $\omega$ is non-zero, as required.   The Ore table for $(p,e_0,e) = (3,2,3)$ omits $0$ and has first row ``$\cdot \; 2 \; 1$'', 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 $\sigma(i) = i^3 =  -i$;  thus
$\mu_{\rm wild}$ satisfies the 
$\sigma$-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 {\em 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 {\em tame conductor $c_t$}, the sum 
of the $e_i-1$.  Third, particular to our wild situation, one has the
{\em wild conductor $c_w$}, the sum of the first subscripts $s_i$.  
Thus for the wild partition \eqref{examplewild}, one has
$(n,c_t,c_w) = (29,21,22)$.     


\section{The generating functions $\Phi_{\cF}(x,y,z)$ and $\Lambda_{\cF}(x,y,z)$}
\label{Three}
      
      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 $\cF = (p,e_0,f_0)$.   When we are continuing the example started in \eqref{exampleordinary}, \eqref{examplewild}, we will take $\cF = (3,2,1)$.  

\paragraph*{Irreducible and isotypical partitions.}
      We say a wild partition is {\em irreducible} if it is non-zero 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, $\mu_{\rm wild}$ has seven irreducible constituents,
\begin{equation}
\label{examplewild2}
\mu_{\rm wild} = 9_{20; 1,1,0,2,2,0,1} + 7 + (3_{1;i} + 3_{1;-i})+ 2 + 2 + 2 + 1.
\end{equation}
We similarly say that a wild partition is {\em isotypical} if it has the form $m \mu$ for
$\mu$ an irreducible wild partition and $m$ a positive integer.  Every wild partition is uniquely 
the sum of its isotypical constituents.  For example,  $\mu_{\rm wild}$ 
has five isotypical constituents,
\begin{equation}
\label{examplewild3}
\mu_{\rm wild} = 9_{20; 1,1,0,2,2,0,1} + 7 + (3_{1;i} + 3_{1;-i})+ (2 + 2 + 2)+ 1.
\end{equation}
If the wild partition $\mu$ is irreducible, we say that the isotypical wild partition 
$m \mu$ has mass $1/m$.     In the number-theoretic settings of  
Sections~\ref{Eisenstein}-\ref{adic2}, 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 $1-1/m$.  



\paragraph*{Definition of $\Phi_\cF(x,y,z)$ and $\Lambda_{\cF}(x,y,z)$.}
      Let $\phi_{\cF,n,c_t,c_w}$ be the { total mass} of $\cF$-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,
\begin{eqnarray}
\label{phi1}
\Phi_{\cF}(x,y,z) & = & \sum_{n=0}^\infty \sum_{c_t=0}^\infty \sum_{c_w=0}^\infty \phi_{\cF,n,c_t,c_w} x^n y^{c_t} z^{c_w} \\
\label{phi2}& = & 
\sum_{\mu} \sum_{m=1}^\infty \frac{1}{m} x^{m n(\mu)} y^{m c_t(\mu)} z^{m c_w(\mu)} \\
& = &
\label{phi3} \sum_{\mu} \log \left( \frac{1}{1 - x^{n(\mu)} y^{c_t(\mu)} z^{c_w(\mu)} } \right).
\end{eqnarray}
Similarly, let $\lambda_{\cF,n,c_t,c_w}$
be the { total number} of $\cF$-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, 
\begin{eqnarray}
\label{lam1} \Lambda_{\cF}(x,y,z) & = &
 \sum_{n=0}^\infty \sum_{c_t=0}^\infty \sum_{c_w=0}^\infty 
 \lambda_{\cF,n,c_t,c_w} x^n y^{c_t} z^{c_w} \\
\label{lam2}& = & 
\prod_{\mu} \sum_{m=0}^\infty x^{m n(\mu)} y^{m c_t(\mu)} z^{m c_w(\mu)} \\
\label{lam3} & = &
\prod_{\mu}  \left( \frac{1}{1 - x^{n(\mu)} y^{c_t(\mu)} z^{c_w(\mu)} } \right).
\end{eqnarray}
The presence of $1/m$ in \eqref{phi2} and the absence of a corresponding 
factor in \eqref{lam2} reflects that \eqref{phi1}-\eqref{phi3} are in the setting of total mass while
\eqref{lam1}-\eqref{lam3} are in the setting of total number.


Comparison of \eqref{phi1}-\eqref{phi3} with  \eqref{lam1}-\eqref{lam3} shows that 
one has an exponential formula
\begin{equation}
\label{PhiLambda}
\Lambda_\cF(x,y,z) = \exp \left( \Phi_{\cF}(x,y,z) \right) .
\end{equation}
We will have analogous exponential formulas on the level of fields and algebras
in the sequel.  

\paragraph*{Computation of $\Phi_\cF(x,y,z)$ and $\Lambda_{\cF}(x,y,z)$.}   
  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 {\em ramification index} and $f$ the {\em 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 $c_w = f s$, this equation being 
 $2 = 2 \cdot 1$ in the first case and $0 = 3 \cdot 0$ in the second.  
 
      Let $\phi_\cF(e,f,s)$ be the total mass of isotypical $\cF$-wild partitions with 
ramification index $e$, inertial degree $f$, and Swan conductor $s$.   One 
has 
\begin{equation}
\label{abstractKrasner}
\phi_\cF(e,f,s) = \frac{1}{f} |W(p,e_0,e,s)^{\sigma^f}| = \frac{1}{f} q^{f \, d(p,e_0,e,s)}  \left( 1 - \delta_{s}^{wE} q^{-f} \right).
\end{equation}
Here, as usual, $X^g$ is the set of fixed points of an operator $g$ on a set $X$.   Also, to unify cases,
$\delta_{s}^{wE}$ is $1$ in the non-maximal case $s<wE$ and $0$ in the 
maximal case $s = w E$.  

Since \eqref{abstractKrasner} 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 $\sigma = \sigma_3$ and take also $f = 2$.
 The fixed point set
of $\sigma^2$ on $W(p,e_0,e,s) \cong \overline{\F}_3-\{0\}$ is $\F_9 - \{0\}$.  
The Frobenius 
operator $\sigma$ on $\F_9 - \{0\}$ 
has three orbits of size two, corresponding to the irreducible
wild partitions $3_{1,\omega} + 3_{1,\omega^3}$ for 
$\omega \in \{i,1+i,2+i\}$.   Similarly, $\sigma$ has two orbits of size one, 
corresponding to the isotypical but non-irreducible $\cF$-wild partitions
$3_{1,\omega} + 3_{1,\omega}$ for $\omega \in \{1,2\}$; 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 \eqref{abstractKrasner} are 
all $4$.  

The concepts of ramification index and inertial degree together with the key formula
\eqref{abstractKrasner} let us evaluate our first generating function explicitly:
\begin{align}
\label{eval1}
\Phi_{\cF}(x,y,z) & = \sum_{e=1}^\infty \sum_{s \in {\rm Ore}(p,e_0,e)} \sum_{f=1}^\infty 
\phi_{\cF}(e,f,s) x^{ef} y^{(e-1)f} z^{sf} \\
\label{eval2} & = \sum_{e=1}^\infty \sum_{s \in {\rm Ore}(p,e_0,e)} \sum_{f=1}^\infty \frac{1}{f} q^{d(p,e_0,e,s) f}  \left( 1 - \delta_s^{wE} q^{-f} \right) x^{ef} y^{(e-1)f} z^{sf} \\
\label{eval3} & = \sum_{e=1}^\infty \sum_{s \in {\rm Ore}(p,e_0,e)} \log \left(  
\frac{1 - \delta_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}
\right).
\end{align}
The logarithm appearing in \eqref{eval3} is very welcome because it cancels the exponential 
in \eqref{PhiLambda} to yield the main result of this section: 
\begin{align}
\label{Threeformula}
\Lambda_{\cF}(x,y,z) & =  \prod_{e=1}^\infty \prod_{s \in {\rm Ore}(p,e_0,e)}  \left(  
\frac{1 - \delta_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}
\right).
\end{align}

\section{The generating function $\Lambda_Q(x)$}
\label{One}
     Formula~\eqref{Threeformula} 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.  

 
 \paragraph*{The specialization $(y,z) = (1,1/q)$.}    
 Equality \eqref{serrespec}, namely $\Lambda_{p,e_0,f_0}(x,1,1/q) = \Lambda(x)$, is established by reducing the $e$-factor of $\Lambda_{p,e_0,f_0}(x,1,1/q)$ to the 
$e$-factor of $\Lambda(x)$:
\begin{eqnarray*}
\Lambda_{p,e_0,f_0;e}(x,1,\frac{1}{q})&  = &  \left( \prod_{s \in {\rm Ore}(p,e_0,e)-\{w E\} }
\frac{1 - q^{d(p,e_0,e,s)-1 -s} x^e}{ 1 - q^{d(p,e_0,e,s)-s} x^e } \right)  
\frac{1}{ 1 - q^{d(p,e_0,e,w E)-w E} x^e } \\
& = & 
\left( \prod_{k=0 }^{w E -d(p,e_0,e,w E)-1}
\frac{1 - q^{-k-1} x^e}{ 1 - q^{-k} x^e } \right) \frac{1}{ 1 - q^{d(p,e_0,e,w E)-w E} x^e }  \\
& = & \frac{1}{1-x^e} .
\end{eqnarray*}
Here the first equality is simply a specialization of \eqref{Threeformula}.  
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 $wE - d(p,e_0,e,w E)$, 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 \cite{Se}.  Serre's mass formula was translated into the 
context of algebras by Bhargava \cite{Bh}.  The technique of generating 
functions was first used in this context by Kedlaya \cite{Ke}.  The 
specialization $(y,z) = (1,1/q)$ is the relevant one for the application
to number fields made by Bhargava \cite{Bh}, \cite{Bh4}.  However
to support our Conjecture~\ref{mainconj} below, we need another
specialization.

\paragraph*{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 
$\Lambda_{p,e_0,f_0}(x,1,1)$ by evaluating each $e$-factor separately.  
As for Serre's case, the main point is that $\Lambda_{p,e_0,f_0;e}(x,1,1)$
is given by a telescoping product.   In our case, however, the cancellation is not as 
complete.  


\begin{theorem}  
\label{etheorem} 
The quantity $\Lambda_{p,e_0,f_0; e}(x,1,1)$ depends only
on $Q = p^{e_0f_0}$ and is equal to 
\begin{equation}
\Lambda_{Q;e}(x) =
\label{wildformula}  
 \frac{ {
  \prod_{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}}} .
\end{equation}
\end{theorem}

\proof  Let $D(k) = d(p,e_0,e,kE-1)$ be the number of integers in $\{0,\dots,k E-1\}$ which are not in
$\Ore(p,e_0,e)$.     This quantity is best understood by thinking in terms of the $j^{\rm th}$ shifted block
$[(j-1) E,j E - 1]$ rather that the $j^{\rm th}$ block $[(j-1) E+1,j E]$.  
In these terms, $D(k)$ is the number of omitted entries in the
first $k$ shifted blocks of the corresponding Ore table.
%  Always the $j^{\rm th}$ shifted  block starts with
%the omission $p^{(j-1) E}-1$ and thereafter omits every $p^{j{\, \rm th}}$ integer.
The $j^{\rm th}$ block contains exactly those integers in $[(j-1) E,j E - 1]$ which are 
not multiples of $p^j$.   
So 
\begin{equation}
\label{Deval}
D(k) =   E \sum_{j=1}^k p^{-j}  = E \frac{p^{-1}-p^{-k-1}}{1-p^{-1}} = 
e_0 t p^w \frac{1-p^{-k}}{p-1} = e_0 t \frac{p^w-p^{w-k}}{p-1}.
\end{equation}
The $e$-factor  $\Lambda_{p,e_0,f_0;e}(x,1,1)$ then simplifies to
the $e$-factor of $\Lambda_{Q}(x)$ as follows:
\begin{eqnarray*}
\Lambda_{p,e_0,f_0;e}(x,1,1) & = & \left( \prod_{s \in {\rm 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} \right)
\frac{1 }{1 - q^{d(p,e_0,e,wE)} x^e} \\
&=&\left(
\prod_{j=1}^{w} \prod_{d = D(j-1)+1}^{D(j)} \left( \frac{1 - q^{d-1} x^e}{1 - q^d x^e} \right)^{p^{j}-1}
\right)
\frac{1 }{1 - q^{D(w)} x^e} \\
& = & \left( \prod_{j=1}^{w} \left( \frac{1 - q^{D(j-1)} x^e}{1 - q^{D(j) }x^e} \right)^{p^{j}-1} \right) 
\frac{1 }{1 - q^{D(w)} x^e} \\
& = & \frac{\prod_{j=1}^{w} (1 - q^{D(j-1)} x^e)^{(p-1) p^{j-1}}}{(1 - q^{D(w)} x^e)^{p^w}} \\
& = & 
\frac{ { \prod_{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}}} .
\end{eqnarray*}
Here the first equality is simply a specialization of \eqref{Threeformula}.  
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 $k^{\rm th}$ 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
\eqref{Deval} and the fact that $q^{e_0} = Q$.   The statement of the theorem
is then obtained by the index shift $j \mapsto j-1$.  (The indexing in the 
statement is preferable in the sequel; otherwise, e.g., the three $p^j$ in 
\eqref{globalminus} would all have to be $p^{j-1}$.)  
 \qed
 
  For wildness $w$ in $\{0,1,2,3\}$, Formula~\eqref{wildformula} simplifies to 
\begin{eqnarray*}
\Lambda_{Q;t}(x) & = & \frac{1}{1-x}, \\
\Lambda_{Q;pt}(x) & = & \frac{(1-x^e)^{p-1}}{(1 - Q^t x^e)^p}, \\
\Lambda_{Q;p^2 t}(x) & = & \frac{(1-x^e)^{p-1} (1 - Q^{p t} x^e)^{p^2-p}}{(1 - Q^{(p+1) t} x^e)^{p^2}} , \\
\Lambda_{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}}.
\end{eqnarray*}
These cases more than suffice for the factors of $\Lambda_2(x)$ and
$\Lambda_3(x)$ displayed in the introduction.   In general, to pass from 
$\Lambda_{p;e}(x)$ to $\Lambda_{p^{n_0};e}(x)$, each written in this form,
one simply replaces each factor of the form $1-c x^e$ by a new factor $1 - c^{n_0} x^e$.  


       For $Q$ regarded as a formal variable, the right side of \eqref{wildformula} 
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 $\geq 1$, so 
that $Q = p^\nu$ for some $\nu \geq 0$.  As remarked already
in the introduction, $Q$ no longer determines
$p$ in our enlarged context.  Accordingly, we write $\Lambda_{p,Q}(x)$ rather than $\Lambda_p(x)$.  

   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 \eqref{wildformula} reduce to $1-x^e$, with $p^w-1$ such 
factors on the top and $p^w$ on the bottom.  Thus \eqref{wildformula} reduces
to $1/(1-x^e)$ independent of the wildness $w$, and so
 $\Lambda_{p,1}(x) = \Lambda(x)$ for all $p$.   This observation further justifies
 our terminology ``wild partitions,'' because
 wild partitions are now continuously related to ordinary partitions.  
 



 \section{Relation with $p$-cores}
 \label{Core} 
     In this section and the next, we continue with the set-up just introduced, so that
$Q \geq 1$ 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 $e = t p^w$ 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.  
 
\paragraph*{$p$-cores.}
      Define 
 \begin{align}
 \label{pcore}
 \Theta_p(x) & = \frac{\Lambda(x)}{\Lambda(x^p)^p}  =
\left(  \prod_{{\rm ord}_p(e) \geq 1} (1-x^e)^{p-1}  \right) \left( \prod_{{\rm ord}_p(e) = 0} \frac{1}{1-x^e} \right).
 \end{align}
  The series $\Theta_p(x)$ is the generating function
   for the sequence of $p$-cores,  i.e.\ partitions without
 hook-lengths a multiple of $p$, as studied in \cite{GKS}.  In the cases $p=2$ and $3$, respectively,
 $\Theta_p(x)$ is the generating function for \seqnum{A010054} and \seqnum{A033687}:
 \begin{align*}
 \Theta_2(x) & =  \sum_{j=-\infty}^\infty x^{2 j^2 - j} &&  =  1 + x + x^3 + x^6 + x^{10} + x^{15} + x^{21} + 
 x^{28} +   \cdots \\
 \Theta_3(x) & =  \sum_{j=-\infty}^\infty   \sum_{k=-\infty}^\infty     x^{3(j^2 + j k + k^2) -j-2 k}                                              && =   1 + x + 2 x^2 + 2 x^4 + x^5 + 2 x^6 + x^8 + 2 x^9 + \cdots 
 \end{align*}
 The reference \cite{GKS} similarly identifies $\Theta_p(x)$ 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 $\Theta_p(x)$ has non-negative coefficients, something not obvious from \eqref{pcore}.  
 


 \begin{figure}[htb]
 \begin{center}
 \epsfig{file=thetaboth.eps,width=6in}
%\includegraphics[width=6in]{thetaboth}
\parbox{5in}{
\caption{\label{contoursC} Contour plots of $|\Theta_2(x)|$ on the left and $|\Theta_3(x)|$ 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~\ref{boundvalues}. They give 
the main contribution to the numbers in Table~\ref{cvalues}.     }}
\end{center}
 \end{figure}


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

\paragraph*{$\Lambda_{p,Q}(x)$ as a product over $p$-cores.}
 
 \begin{corollary}  One has the identity 
  \begin{eqnarray}
\label{globalminus} \Lambda_{p,Q}(x) & = & \prod_{j=0}^\infty \Theta_p(Q^{(p^{j}-1)/(p-1)} x^{p^{j}})^{p^{j}}.
\end{eqnarray}
\end{corollary}

\noindent Before beginning the general proof, it is worth noting that the right side of \eqref{globalminus} 
in the case of $Q = 1$ is a telescoping product, namely
\begin{equation}
\Theta_p(x) \Theta_p(x^p)^p \Theta_p(x^{p^2})^{p^2} \cdots  = \frac{\Lambda(x)}{\Lambda(x^p)^p}
\frac{\Lambda(x^p)^p}{\Lambda(x^{p^2})^{p^2}}
\frac{\Lambda(x^{p^2})^{p^2}}{\Lambda(x^{p^3})^{p^3}} \cdots = \Lambda(x).
\end{equation}
This remark establishes  \eqref{globalminus} in the case $Q = 1$, as we have
already remarked at the end of the previous section that $\Lambda_{p,1}(x) = \Lambda(x)$.  


\proof  Substituting $Q^{(p^{j}-1)/(p-1)} x^{p^{j}}$ in for $x$ in \eqref{pcore}, 
we get that the  factor on the right
of \eqref{globalminus}  with index $j$ is
\begin{eqnarray}
\nonumber \lefteqn{ \Theta_p(Q^{(p^{j}-1)/(p-1)} x^{p^{j}})^{p^{j}}  } \\
\nonumber  &=& \left(  \prod_{{\rm ord}_p(e) > 0} (1-Q^{(p^{j}-1)e/(p-1)} x^{p^{j} e})
  \right)^{p^{j+1}-p^{j}}
\left( \prod_{{\rm ord}_p(e) = 0} \frac{1}{1-Q^{(p^{j}-1)e/(p-1)} x^{p^{j} e}} \right)^{p^{j}} \\
\label{rightefactor} &=& \left(  \prod_{{\rm ord}_p(e) > j} (1-Q^{(1-p^{-j})e/(p-1)} x^e) 
\right)^{p^{j+1}-p^{j}} 
\left( \prod_{{\rm ord}_p(e) = j} \frac{1}{1-Q^{(1-p^{-j})e/(p-1)} x^e} \right)^{p^{j}} 
\end{eqnarray}
Let $e = p^w t$ as usual.  Then 
the $e$-factor of the left side of \eqref{globalminus} is given by \eqref{wildformula} and 
the $e$-factor of the right side of \eqref{globalminus} is the product of \eqref{rightefactor} for
$j = 0$, \dots, $w$.   The $w$ numerator factors of \eqref{wildformula} match the
numerator factors in \eqref{rightefactor} for $j = 0, \dots, w-1$.  The denominator
factor of \eqref{wildformula} matches the denominator factor in \eqref{rightefactor} for 
$j=w$.  
 \qed



\section{Asymptotics}
\label{Asymptotics}  
   In this section we study the asymptotic behavior
of the coefficients $\lambda_{p,Q,n}$ of the power series $\Lambda_{p,Q}(x)$.    

\paragraph*{A change of variables.}   Abbreviate $Q^{-1/(p-1)}$ by $r \in (0,1]$ and change
variables via $x = ry$.   Define $\underline{\Lambda}_{p,Q}(y) = \Lambda_{p,Q}(r y) = \sum_{n=0}^\infty \underline{\lambda}_{p,Q,n} y^n$ so that $\underline{\lambda}_{p,Q,n} =  r^n \lambda_{p,Q,n}$.  
 Then \eqref{globalminus} takes on the simpler form
 \begin{equation}
 \label{globalminus2}
 \underline{\Lambda}_{p,Q}(y) = \prod_{j=0}^\infty \Theta_p(r y^{p^{j}})^{p^{j}}.
 \end{equation}
 As the radius of convergence of $\Theta_p(x)$ is $1$, the radius of convergence of 
 the $j^{\rm th}$ factor is $r^{-1/p^{j}}$.   So as $j$ increases from $1$, these radii decrease
monotonically from $1/r$ with limit $1$.  

\begin{figure}[tbh]
\begin{center}
\epsfig{file = cascades.eps,width=6in}
%\includegraphics[width=6in]{cascades}
\parbox{5in}{\caption{\label{cascades}
{Points $(n,\log(\underline{\lambda}_{2,2,n; k}))$ on the left 
and $(n,\log( \underline{\lambda}_{3,3,n; k}))$ on the right, with $k$ indicated by text.}} }
\end{center}
\end{figure}
    We indicate with a subscript $k$ the corresponding objects where the product in \eqref{globalminus2} is taken from $0$ to $k$.   For any given $k$, the coefficients $\underline{\lambda}_{p,Q,n;k}$ eventually
decay exponentially, by the above radius of convergence remarks.  Figure~\ref{cascades} illustrates
the decay.  


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

 \paragraph*{Root growth.}    Sharpening the statement $\limsup \lambda_{p,Q,n}^{1/n} = Q^{1/(p-1)}$ just observed is the following.
 \begin{proposition} \label{rootprop}
 ${\displaystyle \lim_{n \rightarrow \infty} \lambda_{p,Q,n}^{1/n} = Q^{1/(p-1)}}$.
 \end{proposition}
 \proof  We need only show that the sequence $\lambda_{p,Q,n}^{1/n}$ has no limit points 
 smaller than $Q^{1/(p-1)}$.  In other words, we need only show that the sequence 
 $\underline{\lambda}_{p,Q,n}^{1/n}$ has no limit points smaller than $1$.    From
 \eqref{pcore}, we know that the expansion of $\Theta_p(x)$ begins with 
 $\sum_{n=0}^{p-1} \lambda_n x^n$ which is term-by-term at least 
 $\sum_{n=0}^{p-1} x^n$.    The $j^{\rm th}$ factor $\Theta_p(r y^p)^{p^j}$ of \eqref{globalminus}
  is coefficient-wise bounded from below by $\Theta_p(r y^p)$ which is in turn 
 coefficient-wise bounded from below by $\sum_{a=0}^{p-1} r^a y^{a p^j}$.   So
 $\Lambda_{p,Q}(y)$ is coefficient-wise bounded from below by 
 $\prod_{j=0}^\infty \sum_{a=0}^{p-1} r^a y^{a p^j}$.    If $n = \sum_{i=0}^{\log_p(n)} a_i p^i$ with 
 $0 \leq a_i \leq p-1$, then 
 \begin{equation*}
 \underline{\lambda}_{p,Q,n}  \geq   \prod_{i=0}^{\log_p(n)} r^{a_i} 
   \geq  r^{(1 + \log_p(n))(p-1)}  
 \end{equation*}
 so that 
  \begin{equation}
  \label{rootlowerbound}
 \underline{\lambda}^{1/n}_{p,Q,n}  \geq   
    r^{(1 + \log_p(n))(p-1)/n}.  
   \end{equation}
The right side of \eqref{rootlowerbound} tends to $1$ with $n$, proving that indeed  
$\underline{\lambda}_{p,Q,n}^{1/n}$ has no limit points smaller than $1$. 
 \qed


\paragraph*{Expected refined asymptotics.}
Proposition~\ref{rootprop} is more than we need to support 
Conjecture~\ref{mainconj} 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 \eqref{asymphope2}, conjecturally extending the Hardy-Ramanujan asymptotic for 
$\lambda_{n}$.  

     Another model of the type of statement sought
 is given in the next section for analogous quantities $\lambda_{F,n}$
for $F = \R$ and $F = \C$.   There, \eqref{ArchAsymp2} gives an asymptotic equivalent to 
the root decay factor $\lambda_{F,n}^{1/n}$, analogous to our Proposition~\ref{rootprop}.
More subtly, \eqref{ArchAsymp2} also says
that the ratio decay factor $\lambda_{F,n}/\lambda_{F,n-1}$ 
has the same asymptotic equivalent.  Finally \eqref{ArchAsymp1} is the sharpest 
statement, giving asymptotic equivalents to $\lambda_{F,n}$ itself.   

  
 \paragraph*{Ratio oscillation.}
 In contrast to both ordinary partitions and the Archimedean cases, evidence strongly suggests that 
 $\lambda_{p,Q,n}/\lambda_{p,Q,n-1}$ does not have 
 a limiting value for $Q > 1$. 
  Figure~\ref{startgrowth} graphs $\log(\underline{{\lambda}}_{p,Q,n})$ for
 \begin{figure}[tbh]
\begin{center}
\epsfig{file = twostarts.eps,width=6in}
%\includegraphics[width=6in]{twostarts} 
\parbox{5in}{\caption{\label{startgrowth}  Points $(n,\log(\underline{\lambda}_{2,2,n}))$ on the left and 
$(n,\log(\underline{\lambda}_{3,3,n}))$ on the right, for $0 \leq n \leq 100$.  Also  
points $(n,\log(\underline{\lambda}_{2,4,n}))$ on the left and 
$(n,\log(\underline{\lambda}_{3,9,n}))$ on the right, for $36 \leq n \leq 100$.  The oscillatory 
behavior modulo eight on the left and modulo nine on the right matches Table~\ref{cvalues} well.}}
\end{center}    
\end{figure}
 $p=2,3$ and $Q=p^j$ with 
 $j=1,2$.   One sees smooth growth 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.  
 
  \begin{figure}[tbh]
\begin{center}
\epsfig{file=twolambdas.eps, width=6in}
%\includegraphics[width=6in]{twolambdas}
\parbox{5in}{\caption{\label{lambdacontours} Contour plots of $|\underline{\Lambda}_{2,2}(y)|$ on the left and 
$|\Lambda_{3,3}(y)|$ on the right, 
both on the open unit disk $|y|<1$, with dark shading indicating large values.  }}
\end{center}
\end{figure}


     Figure~\ref{lambdacontours} illustrates with two examples 
     the magnitude of $\underline{\Lambda}_{p,Q}(y)$ 
 as a function of the complex variable $y$.  The needed Fourier analysis is connected with the
 limiting behavior of $\underline{\Lambda}_{p,Q}(y)$ as $|y|$ increases to $1$.  
For $r<1$, consider the function
\begin{equation}
\label{chat}
\hat{c}_{p,Q}(y) = \prod_{j=0}^\infty
 \left( \frac{\Theta_p(r y^{p^j} )}{\Theta_p(r)} \right)^{p^j}
\end{equation}
defined on the unit circle.  
 One has $\hat{c}_{p,Q}(1) = 1$ 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
 $0 < |\hat{c}_{p,Q}(y)| < 1$ for these $y$.  Finally if $y$ is otherwise, the infinite product 
 converges to $0$. 
 
 We view $\hat{c}_{p,Q}(y)$ as giving the normalized boundary values of 
 $\underline{\Lambda}_{p,Q}(y)$. Intuitively, we can view $\underline{\Lambda}_{p,Q}(y)$ 
 as having its most
 important singularity at $1$.   This singularity is echoed in quantitatively smaller
 singularities at primitive $p^{\rm th}$ 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~\ref{lambdacontours}.      Table~\ref{boundvalues} 
 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 $Q$.  
 \begin{table}[tbh]
 \[
\begin{array}{|lll|c|lll|}
\cline{1-3} \cline{5-7}
\alpha & \hat{c}_{2,2}(e^{2 \pi i \alpha}) &
\hat{c}_{2,4}(e^{2 \pi i \alpha})  & & \alpha &
\hat{c}_{3,3}(e^{2 \pi i \alpha})  &
\hat{c}_{3,9}(e^{2 \pi i \alpha})  \\
\cline{1-3} \cline{5-7}
0/1 &    1.               &1.
            & \;\;\;\;\;\; &  0/1  &1.               &1.               \\
1/8 &    0.0005 + 0.0009 i&0.0567 + 0.0405 i&& 1/9 & 0.0004 + 0.0010 i &0.0542 + 0
.0569 i\\
1/4 &    0.0341 + 0.0130 i &0.2660 + 0.0624 i &&2/9&0.0006 + 0.0002 i&0.0560 - 0.0
029 i \\
3/8 &    0.0007 - 0.0001 i&0.0490 - 0.0119 i &&1/3&0.1191 + 0.0210 i &0.4476 + 0.0
718 i\\
1/2 &    0.2385           &0.5803           &&4/9&0.0007 + 0.0001 i&0.0463 + 0.020
5 i\\
\cline{1-3} \cline{5-7}
\end{array}
\]
\begin{center}
\parbox{5in}{\caption{\label{boundvalues}   Some normalized boundary values $\hat{c}_{p,Q}(y)$ of 
$\underline{\Lambda}_{p,Q}(y)$, rounded to the nearest ten-thousandth.  Only values in 
the upper half plane are given, because $\hat{c}_{p,Q}(\bar{y}) = \overline{\hat{c}_{p,Q}(y)}$.}}
\end{center}
\end{table}

  








 
          The function $\hat{c}_{p,Q}(y)$  enters into our 
Fourier analysis as follows.    Taking an inverse Fourier transform, define
\begin{equation}
\label{inversetransform}
c_{p,Q}(n) = \sum_{y} y^{-n} \hat{c}_{p,Q}(y),
\end{equation}
the sum being over all $p^{\rm th}$ power roots of unity.
One can check that the sum in \eqref{inversetransform} 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~\ref{cvalues} gives some values.
\begin{table}[tbh]
\[
\begin{array}{|rrr|c|rrr|}
\cline{1-3} \cline{5-7}
n & c_{2,2}(n) & c_{2,4}(n) &
\;\;\;\;\;\;\;\;\; & n & c_{3,3}(n) & c_{3,9}(n) \\
\cline{1-3} \cline{5-7}
0 & 1.309 & 2.324 && 0 & 1.242 & 2.208 \\
                    1 & 0.788  & 0.596  && 1 & 0.918  & 0.774  \\
                    2 & 1.172 & 1.153 && 2 & 0.847  & 0.496  \\
                    3 & 0.737  & 0.324  && 3 & 1.238 & 1.878 \\
                    4 & 1.304 & 1.901 && 4 & 0.918  & 0.679  \\
                    5 & 0.787  & 0.493  && 5 & 0.845  & 0.426  \\
                    6 & 1.168 & 0.944  && 6 & 1.235 & 1.600 \\
                    7 & 0.734  & 0.265  && 7 & 0.915  & 0.577  \\
\cline{1-3}
\multicolumn{3}{c}{\;}        &  & 8 & 0.842  & 0.362  \\
\cline{5-7}
\end{array}
\]
\begin{center}
\parbox{5in}{\caption{\label{cvalues} 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 $9$.  }}
\end{center}
\end{table}
We expect that the function $c_{p,Q}(n)$ fully captures the oscillatory behavior in the 
sense that  
\begin{equation}
\label{ratiolimit}
\frac{\underline{\lambda}_{p,Q,n}/c_{p,Q}(n)}{\underline{\lambda}_{p,Q,n-1}/c_{p,Q}(n-1)} \sim
 Q^{1/(p-1)}.
\end{equation}
Computations such as those illustrated by Figure~\ref{endgrowth} support this expectation. 


\paragraph*{Towards an asymptotic equivalent to $\lambda_{p,Q,n}$.}  The smooth part of
$\lambda_{p,Q,n}$  presents more of a mystery.  
Computations are consistent with the conjecture given as \eqref{asymphope1} in the introduction, 
namely
\begin{equation}
\label{asymphope2}
\lambda_{p,Q,n} \sim c_{p,Q}(n) C_p(Q) n^{B_p(Q)} e^{A_p(Q) \sqrt{n}} Q^{n/(p-1)}
\end{equation}
for quantities $A_p(Q)$, $B_p(Q)$, and $C_p(Q)$ to be thought of as functions
on the $Q$-interval $[1,\infty)$.    

     For $Q = 1$, the oscillatory factor  $c_{p,Q}(n)$ reduces to $1$.    One has 
\[
(A_p(1), B_p(1), C_p(1))  =  (\pi \sqrt{\frac{2}{3}} , -1, \frac{1}{4 \sqrt{3}}) 
\approx (2.56,-1.00,0.144),
\]
independently of $p$, by the Hardy-Ramanujan asymptotic for partitions \cite{HR}.    
Figure~\ref{endgrowth} 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 $\log(\underline{\lambda}_{p,Q,n}/c_{p,Q}(n))$
over $[20,4000]$.     The drawn lines are thick enough so that they contain all the
actual points $(n,\log(\underline{\lambda}_{p,Q,n}/c_{p,Q}(n)))$.     For the most
important quantity $A_p(Q)$, the fit yields
\begin{align*}
A_2(2) & \approx  1.66 \, , & A_3(3) & \approx 1.68 \, ,\\
A_2(4) & \approx  1.18 \, , & A_3(9) & \approx 1.21 \, .
\end{align*}
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.  



\begin{figure}[tbh]
\begin{center}
\epsfig{file=endgrowth.eps,width=6in}
%\includegraphics[width=6in]{endgrowth3}
\parbox{5in}{
\caption{\label{endgrowth}  Least square fits to points 
$(n,\log(\underline{\lambda}_{2,2^j,n}/c_{2,2^j}(n)))$ on the left and 
$(n,\log(\underline{\lambda}_{3,3^j,n}/c_{3,3^j}(n)))$ on the right, for $j \in \{1,2\}$ and 
$20 \leq n \leq 4000$.}  }  
\end{center}
\end{figure}




\section{Fields, algebras, and the cases $F = \R$ and $F = \C$}
\label{RC}
         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~\ref{NumberFields}.    
         
         
\paragraph*{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 $\phi_{F,n}$ and $\lambda_{F,n}$ to be the total masses of 
 $\Fields_{F,n}$ and $\Algebras_{F,n}$ 
respectively.    Let 
\begin{align*}
\Phi_{F}(x) & = \sum_{n=1}^\infty \phi_{F,n} x^n  = x + \cdots, &
\Lambda_{F}(x) & = \sum_{n=0}^\infty \lambda_{F,n} x^n  = 1 + x + \cdots 
\end{align*}
be the corresponding generating functions.  Then one has the exponential formula \cite[Chapter 5]{St}
\begin{equation}
\label{expformula}
\Lambda_{F}(x) = \exp(\Phi_{F}(x)),
\end{equation}
from the definition of mass and the way algebras are built from fields.   
         
\paragraph*{The cases $F=\R$ and $F=\C$.}          With the above definitions, $\Fields_{\R,1} = \{\R\}$, $\Fields_{\R,2} = \{\C\}$, and otherwise
 $\Fields_{\R,n} = \emptyset$.     Also $\Algebras_{\R,n} = \{\R^{r} \C^s : r + 2 s = n\}$, 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
 \begin{align}
 \label{functR}
 \Lambda_\R(x) & = \sum_{n=0}^\infty \lambda_{\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 +\cdots ,  \\
 \label{functC}
 \Lambda_\C(x) & = \sum_{n=0}^\infty \lambda_{\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 + \cdots  .
 \end{align}
 The numbers $n! \lambda_{\R,n}$ form the sequence \seqnum{A000085} giving, among 
 other interpretations, the number of involutions in the symmetric group $S_n$.    One has
 the asymptotic formulas
 \begin{align}
 \label{ArchAsymp1}
 \lambda_{\R,n} & \sim   \frac{e^{\frac{n}{2}+\sqrt{n}-\frac{1}{4}} n^{-\frac{n}{2}-\frac{1}{2}}}{2
    \sqrt{\pi }}  ,
 & \lambda_{\C,n} & \sim  \frac{e^{n}}{ n^{n+\frac{1}{2}} \sqrt{2 \pi }},
 \end{align}
 the first due to Moser and Wyman \cite{MW} and the second being Stirling's approximation.  
   On the level of ratio and
root behavior, one has
 \begin{align}
 \label{ArchAsymp2}
\frac{\lambda_{\R,n}}{\lambda_{\R,n-1}} \sim  \lambda_{\R,n}^{1/n} & \sim  \sqrt{\frac{e}{n}}, &
\frac{\lambda_{\C,n}}{\lambda_{\C,n-1}} \sim   \lambda_{\C,n}^{1/n} & \sim  \frac{e}{n},
 \end{align}
 thus superexponential decay.  


 

\section{Eisenstein polynomials}
\label{Eisenstein}  
           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~\ref{Asymptotics}.  
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.\ \cite{Go}.     We need mainly the algebraic theory of 
finite degree field extensions of $\Q_p$, i.e.\  Chapter~5 of \cite{Go}.  

\paragraph*{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 $\cO$ be the ring of integers of $F$,
let $\Pi$ be its maximal ideal, and let $\kappa = \cO/\Pi$.  
The ramification index of $F/\Q_p$ is the positive integer $e_0$ such that $\Pi^{e_0} = (p)$.
 The inertial degree of $F/\Q_p$ is
the positive integer $f_0$ such that $q := |\kappa| = p^{f_0}$.   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 
$\pi$ of $F$, i.e.\ a generator of $\Pi$.    For $a \in \cO - \{0\}$, we write $\ord_{\pi}(a) = b$
to mean that $a$ generates the ideal $(\pi^b)$.     We define $\ord_\pi$ on
all of $\cO$ by writing $\ord_\pi(0) = \infty$.  

\paragraph*{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 $\Pi^c$ in $\cO$.    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 
\begin{equation}
\label{discform}
D(g) =  (-1)^{n(n-1)/2} \Res_x(g(x),g'(x))
\end{equation}
 has the form $\Pi^{c + 2d}$ 
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 $f s$, where $s \in \Ore(p,e_0,e)$ 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 
 \begin{equation}
 \Fields_{F,n} = \coprod_{ef=n} \coprod_{s \in {\rm Ore}(p,e_0,e)} \Fields_{F}(e,f,s) 
 \end{equation}
 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.   
 
 
 \paragraph*{Eisenstein polynomials.}
   Consider monic polynomials of degree
$e$ with coefficients in $\cO$.  Such a polynomial  
\begin{equation}
\label{eispoly}
g(x) = x^e + a_{e-1} x^{e-1} + \cdots + a_{1} x + a_0
\end{equation}
is called an Eisenstein polynomial if and only if $\pi$ divides all
the coefficients $a_i$ and moreover $\pi^2$ does not divide $a_0$.  
Let $\Eis(\cO,e)$ be the space 
of degree $e$ Eisenstein polynomials over $\cO$.

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



   Conversely, suppose a totally ramified $K$ is given.  Then one can consider 
for each of its uniformizers $\omega$  the characteristic polynomial
$g_\omega(x)$ of $\omega$ acting by multiplication on $K$, where
$K$ is considered as an $e$-dimensional vector space over $F$.  
The resulting map
\begin{equation}
\label{keycover}
\omega \mapsto g_\omega(x)
\end{equation}
is $|\Aut(K/F)|$-to-$1$ over its image 
$\Eis(\cO,e)_K \subseteq \Eis(\cO,e)$.  

     



\paragraph*{Conductors of Eisenstein polynomials.}

The conductor $c = c_t + c_w$ of the Eisenstein polynomial \eqref{eispoly} is
$\ord_x(g'(x))$, where $x$ here is understood as the given uniformizer of 
$\cO[x]/g(x)$  \cite[III.6] {SeCL}.    The tame conductor $c_t$ is $e-1$.   
Thus 
\begin{equation}
\label{eiscond}
c_w = \ord_x(g'(x)) - (e-1).
\end{equation}
 For $i=1$, \dots, $e$, define the $i^{\rm th}$ index 
 of an Eisenstein polynomial \eqref{eispoly} to be
\[
\ind_i(g(x)) =  \ord_x(i a_i x^{i-1}) - (e-1) = e \; \ord_\pi(i a_i) + i - e  
= \left\{
\begin{array}{ll}
{\displaystyle e \;  \ord_\pi(i \frac{a_i}{\pi}) + i ,}& \mbox{if $i<e$,} \\
&\\
e e_0 w =  w E, & \mbox{if $i = e$.} 
 \end{array}
 \right.
\]
One has $\ind_i(g(x)) \equiv i$ 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 $\ord_\pi(a_i)^{\rm th}$ number in 
column $i$ of the corresponding Ore array 
$\Ore(p,e_0,e)$.    Display \eqref{oresecond}, a copy of \eqref{orefirst} except that
Column $i$ has been headed by the corresponding coefficient
$a_i$, illustrates this viewpoint.  
\begin{equation}
\label{oresecond}
\begin{array}{ccccccccc}
a_9 & a_8 & a_7 & a_6 & a_5 & a_4 & a_3 & a_2 & a_1  \\
. &   &    &      &     &   &     &    \\
\hline
. & 8 & 7 & .  & 5 & 4 & .  & 2 & 1  \\
. & 17 & 16 & . & 14 & 13 & . & 11 & 10  \\
\hline
. & 26 & 25 & 24 & 23 & 22 & 21 & 20 & 19  \\
36 & 35 & 34 & 33 & 32 & 31 & 30 & 29 & 28  \\
\end{array}
\end{equation}
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:
\begin{equation}
\label{conditions}
\begin{array}{lcrcr}
\ord_\pi(a_1) & = & 1 & \;\;\;\;\;\;\;\; & 1 \\
\ord_\pi(a_2) & = & 1 &  & 2 \\
\ord_\pi(a_4) & = & 1 &  & 4 \\
\ord_\pi(a_5) & = & 1 &  & 5 \\
                         & \vdots & &  &\vdots \\
\ord_\pi(a_6) & = & 2 &  & 33 \\
\ord_\pi(a_7) & = & 4 &  & 34 \\
\ord_\pi(a_8) & =& 4 &  & 35 \\
\ord_\pi(a_9) & = & 0   &   &36
\end{array}
\end{equation}
One has a natural
decomposition
\begin{align}
\label{eisdecomp}
\Eis(\cO,e) & = \coprod_{s \in {\rm Ore}(p,e_0,e)} \Eis(\cO,e,s) \\
& = \coprod_{s \in {\rm Ore}(p,e_0,e)} \coprod_{K \in {\rm Fields}_F(e,1,s)} \Eis(\cO,e,s)_K
\end{align}
where $\Eis(\cO,e,s)$ consists of all Eisenstein polynomials over
$\cO$ of degree $e$ and wild conductor $s$ and
$\Eis(\cO,e,s)_K$ is the subset consisting of polynomials which define $K$.  

   

\section{$p$-adic algebras: masses}
\label{adic1}
     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 $\phi_{F,n,c_t,c_w}$ 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  $\lambda_{F,n,c_t,c_w}$ be its total mass.    One has the corresponding 
generating functions, satisfying
$\Lambda_{F}(x,y,z) = \exp(\Phi_{F}(x,y,z))$.

     Let $\cF = (p,e_0,f_0)$ be the invariants of $F$.  
This section explains  how the equality
\begin{equation}
\label{FcF}
\Phi_{F}(x,y,z) = \Phi_{\cF}(x,y,z)
\end{equation}
follows from the Krasner mass formula.    It is this equality
which renders the directly defined power series 
$\Phi_{\cF}(x,y,z)$ of interest in algebraic number theory.   
By exponentiating both sides of \eqref{FcF} one immediately gets
$\Lambda_{F}(x,y,z) = \Lambda_{\cF}(x,y,z)$. 
 Besides our
given $F = F_1$, there may be non-isomorphic $F_2$, \dots, $F_m$ with the same
 invariants $(p,e_0,f_0)$.  
 Our notation encourages one to also think of $\cF$ as representing
 the numerical equivalence class $\{F_1,\dots,F_m\}$.   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.  


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

        As in Section~\ref{Wild}, for $s \in \Ore(p,e_0,e)$ let $d(p,e_0,e,s)$ be the 
number of integers in $\{0,\dots, s\}$ which are not in $\Ore(p,e_0,e)$.   
As illustrated by \eqref{conditions}, for a random polynomial to be in $\Eis(\cO,e,s)$ it has to fail 
$s - d(p,e_0,e,s)$ successive tests, each of which is failed with
probability $1/q$.  If $s < w E$, then it moreover has to pass the next test.    
Accordingly,  
\begin{equation}
\label{totalvolume}
\mbox{volume}(\Eis(\cO,e,s)) = q^{-e - s + d(p,e_0,e,s)} (1-q^{-1}) (1 -  \delta_{s}^{wE} q^{-1}),
\end{equation}
with $\delta_{s}^{wE}$ either $1$ or $0$ according to whether $s < wE$ or $s=wE$, as
in Section~\ref{Three}.




\paragraph*{The Krasner mass formula and proof of $\eqref{FcF}$.}  
For $K \in  \Fields_F(e,1,s)$,  let  $U_K$ be its set of uniformizers and let 
$\Eis(\cO,e,s)_K$ be the set of its defining Eisenstein polynomials.   Consider 
again the  degree $|\Aut(K/F)|$ cover $U_K \rightarrow \Eis(\cO,e,s)_K$ 
of \eqref{keycover}.  By a Jacobian computation \cite{Se}, one has  
\begin{equation}
\label{massvolume}
\mbox{mass}(K) =  q^{s+e}  \frac{\mbox{volume}(\Eis(\cO,e,s)_K)}{1-q^{-1}} .
\end{equation}
Summing \eqref{massvolume} over $K \in \Fields_F(e,1,s)$ and eliminating
volumes via \eqref{totalvolume} gives 
\begin{equation}
\label{Krasner}
 \phi_{F}(e,1,s) = 
  q^{d(p,e_0,e,s) } (1 - \delta_{s}^{wE} q^{-1}) .
   \end{equation}
which is  the Krasner mass formula \cite{Kr}. 

    The case of general $f$ reduces to the totally ramified case $f=1$ via
\begin{equation}
\label{basechange1}
\Fields_{F}(e,f,s) = \Fields_{F_f}(e,1,s),
\end{equation}
where $F_f$ is the unique up to isomorphism degree $f$ extension of $F$.   One has
\begin{equation}
\label{basechange2}
\phi_{F}(e,f,s) =   \frac{1}{f} \phi_{F_f}(e,1,s) \\
                     =   \frac{1}{f} q^{f \, d(p,e_0,e,s) } (1 - \delta_{s}^{wE} q^{-f}) .
\end{equation}
Here the first equality of \eqref{basechange2} holds because of
 \eqref{basechange1} and the fact that $|\Aut(F_f/F)| = f$.
The second equality of  \eqref{basechange2} holds because of \eqref{Krasner}, with 
$F$ replaced by $F_f$ on the left and hence $q$ replaced by $q^f$ on the right. 

     The quantities $\phi_{\cF}(e,f,s)$ and $\phi_{F}(e,f,s)$ agree because their 
explicit formulas in \eqref{abstractKrasner} and \eqref{basechange2} agree.   
Replacing $\cF$ by $F$ in the evaluation \eqref{eval1}-\eqref{eval3} of  $\Phi_{\cF}(x,y,z)$ then
shows that $\Phi_{F}(x,y,z)$ evaluates to the same explicit formula.  

 


\section{$p$-adic algebras: geometric packets}  
\label{adic2}



       One obvious difference between the standard $\Lambda_{\R}(x)$, $\Lambda_{\C}(x)$ 
 and our more complicated $\Lambda_Q(x)$ 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 $\Lambda_{p,e_0,f_0}(x,y,z)$, 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~\ref{Wild} and \ref{Three} numerically, this
section goes farther and justifies our combinatorial definitions set-theoretically, explaining
the natural objects to which wild partitions correspond.  

\paragraph*{Algebras over $F^{\rm un}$.}  For the new perspective, we fix 
a maximal unramified extension $F^{\rm un}/F$, with ring of integers $\cO^{\rm un}$ and
associated residual extension
$\overline{\kappa}/\kappa$.  We let $\sigma \in \Gal(\overline{\kappa}/\kappa) =  \Gal(F^{\rm un}/F)$ be the Frobenius element.   

The theory of Eisenstein series over $\cO$ goes through without change over $\cO^{\rm un}$.
Accordingly, the important set  $\Fields_{F^{\rm un},e}$ of degree $e$ field extensions
of $F^{\rm un}$ is identified with the quotient of $\Eis(\cO^{\rm un},e)$ 
modulo an equivalence relation $\sim$, where $g_1(x) \sim g_2(x)$ if and only
if $F^{\rm un}[x]/g_1(x)$ and $F^{\rm un}[x]/g_2(x)$ are isomorphic.  One likewise has
$\Fields_{F^{\rm un}}(e,s) = \Eis(\cO^{\rm un},e,s)/\sim$,
these sets being non-empty exactly for $s \in \Ore(p,e_0,e)$.    Thus
\begin{equation}
\label{Fieldequate}
\Fields_{F^{\rm un}} = \coprod_{e = 1}^\infty \coprod_{s \in {\rm Ore}(p,e_0,e)} \Fields_{F^{\rm un}}(e,s).
\end{equation}
 The Frobenius element $\sigma$ acts compatibly on both sides of 
\eqref{Fieldequate} by taking $K = F^{\rm un}[x]/\sum a_i x^i$ to $K^\sigma = F^{\rm un}[x]/\sum a_i^\sigma x^i$.
In turn, as for any ground field, $\Algebras_{F^{\rm un}}$ is the free abelian monoid generated
by $\Fields_{F^{\rm un}}$.
 We denote by
 $\Algebras_{F^{\rm un}}^\sigma$ the set of $\sigma$-fixed points, as usual.  

\paragraph*{Geometric packets.}  The mass formulas of the previous section transfer to 
cardinality formulas in our new context as follows.  If $K \in \Algebras_F$, then one has its corresponding base-changed algebra 
$K^{\rm un} 
 \in \Algebras_{F^{\rm un}}^\sigma$.   
Explicitly, if $K = F[x]/g(x)$ then $K^{\rm un} = F^{\rm un}[x]/g(x)$.   
The fiber of the map 
\begin{equation}
\Algebras_F \rightarrow \Algebras_{F^{\rm un}}^\sigma
\end{equation}
above a point $L \in \Algebras_{F^{\rm un}}^\sigma$ 
is the set of all $K \in \Algebras_F$ with $K^{\rm un} 
\cong L$, 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 $L \in \Algebras_{F^{\rm un}}$
 and let $A = \Aut(L/F^{\rm un})$.   Let $\cA = \Aut(L/F)$.   Then
one has a short-exact sequence
\begin{equation}
\label{ses}
 A \hookrightarrow \cA \twoheadrightarrow \Gal(\overline{\kappa}/\kappa).
 \end{equation}    
  Let $A^1$ be the set of preimages of $\sigma$ 
 in $\cA$.   So $A$ acts by conjugation on $A^1$.
Then each $\rho \in A^1$  determines a model $L^\rho/F$ of $L/F^{\rm un}$.  
In fact, $L^\rho$ is just the fixed algebra of the subgroup of $\cA$
generated by $\rho$.    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 $L^\rho$ is determined by the conjugacy class $[\rho]$ of
$\rho$.  Also, 
the automorphism group of $L^\rho/F$ is the subgroup of
$A$ which fixes $\rho$.   Finally, one has always 
\begin{equation}
\label{mass1}
\sum_{\rho} \frac{1}{|\Aut(L^\rho/F)|} =  \sum_{\rho} \frac{|[\rho]|}{|A|} = 1,
\end{equation} 
each sum being over representatives of the conjugacy classes in
$A^1$.  The  last equality in \eqref{mass1} follows because $A^1$ is partitioned 
into the classes $[\rho]$ and $|A| = |A^1|$.


\paragraph*{Wild partitions and $F^{\rm un}$-algebras.}  We have established
\begin{equation}
\left| W(p,e_0,e,s)^{\sigma^m} \right| = \left| \Fields_{F^{\rm un}}(e,s)^{\sigma^m} \right|
\end{equation}
as both sides are $q^{f \, d(p,e_0,e,s)} (1 - \delta_s^{wE} q^{-f})$, the left
side via \eqref{abstractKrasner} and the right side via \eqref{basechange2} and the 
mass one principle.     As all orbits on both sides are finite, this implies
\begin{equation}
\label{bijection}
W(p,e_0,e,s) \cong  \Fields_{F^{\rm un}}(e,s)
\end{equation}
as $\sigma$-sets.     A choice of bijections \eqref{bijection} induces a bijection from
$(p,e_0,f_0)$-wild partitions to $\Algebras_{F^{\rm un}}^\sigma$.     In
particular, $\lambda_{F,n,c_t,c_w}$ is the number of $F^{\rm un}$-algebras
of degree $n$, tame conductor $c_t$, and wild conductor $c_w$; this is the promised
conceptual explanation of the integrality of $\lambda_{F,n,c_t,c_w}$.  


\paragraph*{Explicit bijections}  
For wild partitions to truly index geometric algebras, one would need
to choose explicit $\sigma$-invariant bijections from $W(p,e_0,e,s)$ to 
$\Fields_{F^{\rm un}}(e,s)$ 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 \cite{Am}, which also describes the case
$e=p$ for general $F$.   In our setting, the possible Swan conductors are
 $\Ore(p,1,p) = \{1,\dots, p\}$.  
Always the associated dimension is $d(p,1,p,s) = 1$.   
 If $s < p$ then an explicit $\sigma$-equivariant bijection is
 \begin{eqnarray*}
\label{bijection1}
W(p,1,p,s) = \overline{\F}^\times_p & \rightarrow &  \Fields_{F^{\rm un}}(p,s), \\
           a & \rightarrow & \Q_p^{\rm un}[x]/(x^p + p \tilde{a} x^s + p).
\end{eqnarray*}
For $s = p$, an explicit $\sigma$-equivariant bijection is 
 \begin{eqnarray*}
\label{bijection2}
W(p,1,p,p) = \overline{\F}_p & \rightarrow &  \Fields_{F^{\rm un}}(p,p), \\
           a & \rightarrow & \Q_p^{\rm un}[x]/(x^p + p + p^2 \tilde{a}).
\end{eqnarray*}
In each case, $\tilde{a} \in \Z_p^{\rm un}$ is any lift of $a$.  


\paragraph*{Internal structure of geometric packets.}   Let $L \in \Algebras_{F^{\rm un}}^\sigma$. 
In many cases, the corresponding geometric packet
$\{K_1,\dots,K_g\} \subset \Algebras_F$ 
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 \cite{Am} 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 $F^{\rm un}$.  Then the geometric packet of models 
for $L$ consists of algebras $F_\mu$, where $\mu$ is a partition
of $m$.  Here if  $\mu = \mu_1+\dots+\mu_h$ then 
$F_\mu = F_{\mu_1} \times \cdots \times F_{\mu_h}$, where,
as before, $F_f$ denotes the degree $f$ unramified extension
of $F$.   Then $|\Aut(F_\mu/F)| = \prod_k k^{m_k} m_k!$, where $m_k$ is 
the number of times $k$ appears in $\mu$ and Equation~\eqref{mass1} 
becomes the class equation for the symmetric group $S_m$.  

    When a packet $\{K_1,\dots,K_g\}$ contains a totally ramified field then 
all its elements are totally ramified fields.  The database of local fields \cite{JR} 
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~\ref{Wild}.  More generally suppose a packet $\{K_1,\dots,K_g\}$ 
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~\ref{Wild}.  

\section{Number fields}
\label{NumberFields}
   \paragraph*{The sets $\mbox{\rm $\Fields$}_{F,n,S}$.}  Let $F[x] = \Q[x]/g_0(x)$ be a number field.  Let $\cS(F)$ be its set of places, indexing the set of completions of $F$.  Thus 
   $\cS(\Q) = \{\infty,2,3,5,\dots\}$ and $\cS(F)$ maps surjectively to $\cS(\Q)$.  
     
     For $S \subseteq \cS(F)$, 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  $v \in \cS(F)-S$, and all $w \in \cS(K)$ 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 $n \geq 3$, the second largest
 that $\Gal(K^g/F)$ can be is the alternating group $A_n$.   
 We have a decomposition
 \begin{equation}
 \label{globaldecomp}
 \Fields_{F,n,S} =  \Fields^{\rm sym}_{F,n,S}  \coprod  \Fields^{\rm alt}_{F,n,S}  \coprod  
 \Fields^{\rm small}_{F,n,S}
 \end{equation}
 and also write $\Fields^{\rm big}_{F,n,S}$ to indicate the union of the first two parts.  
 We write also $\Fields^s_{F,S} = \coprod_n \Fields^s_{F,n,S}$ 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 $n \geq 8$, a group-theoretical result of Jordan \cite{Jo} suffices:  the Galois 
 group is big if and only if 
 \begin{equation}
 \label{Jordan}
  \mbox{the degree of $K_w/F_v$ is a prime in $(n/2,n-2)$}
 \end{equation}
 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 \eqref{globaldecomp}
we have $\phi_{F,n,S} = \phi^{\rm big}_{F,n,S}  + \phi^{\rm small}_{F,n,S}$.  Our main concern is
 $\phi^{\rm big}_{F,n,S}$, which is just the cardinality of $\Fields^{\rm big}_{F,n,S}$ 
 when $n \geq 4$.  
 
      For $S$ all of $\cS(F)$, 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 $\Fields_{F,n,\cS(F)}$, so that most fields in $\Fields_{F,n,S}$ would be in 
 $\Fields^{\rm sym}_{F,n,S}$.  This section argues that
 the evidence points in the opposite direction, at least when one fixes
 $S$ and considers all  $n$ simultaneously.     
 
  \paragraph*{Ease of constructing fields in $\mbox{\rm $\Fields$}_{F,S}^{\rm small}$.}  
 One has  $\Fields_{\Q,S} = \{\Q\}$ for $S = \{\}$ or $S = \{\infty\}$.  Otherwise, $\Fields_{\Q,S}$
  is infinite, as it at least contains the real cyclotomic fields $\Q[x]/\Phi^+_{p^k}(x)$ for
  all $p$ in $S$ and all positive $k$.    Since the Galois group of 
  $\Q[x]/\Phi^+_{p^k}(x)$ is abelian, these fields are in  
  $\Fields^{\rm small}_{\Q,S}$ whenever their degree is $\geq 4$.   
 
 There is an elaborate theory for describing the part of $\Fields_{F,S}^{\rm small}$ 
consisting of fields $K$ such that $\Gal(K^g/\Q)$ is solvable \cite{Ko, NSW}.  This theory says 
that as soon as $S$ is large enough, $\Fields_{F,S}^{\rm small}$ is very large indeed.  
For example, let $L$ be a maximal pro-2-extension of $\Q$ ramified only within
$S = \{\infty,2\}$.  Then Markshaitis' theorem \cite[Example 11.18]{Ko} says 
$\Gal(L/\Q)$ is the free pro-2 product of $\Z/2$ and
$\Z_2$.  Accordingly $\phi^{\rm small}_{\Q,2^k,\{\infty,2\}}$ grows exponentially with
$n = 2^k$.  

There are also general techniques for constructing non-solvable fields
in  $\Fields_{F,S}^{\rm small}$.  For example using modular forms gives 
fields with Galois groups with $PSL_2(\F_{\ell^f})$ as a simple
subquotient.    Already this technique shows that the
Galois group corresponding to  any $\Fields_{\Q,\{\infty,p,\ell\}}$ has 
infinitely many simple subquotients different from $A_n$.  
The $ABC$-construction of \cite{Ro} shows that one can likewise
expect infinitely many simple subquotients of the form
$PSp_{2k}(\F_\ell)$ involved in $\Fields_{\Q,S}$ for 
$S$ large enough, e.g.\ $S = \{\infty,2,3\}$.  
        
  
 \paragraph*{Difficulty of constructing  fields in $\mbox{\rm $\Fields$}_{F,S}^{\rm 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 $t \in \Q-\{0,1\}$ consider  the polynomial 
\begin{equation}
g_{n,t}(x) = x^n - n t x + (n-1) t. 
\end{equation} 
Its  discriminant is
\begin{equation}
D_{n,t} = (-1)^{(n-1)(n-2)/2} n^n (n-1)^{n-1} t^{n-1} (t-1).
\end{equation}
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
 \cite[Section~10]{Ro}, with
two relatively prime parameters $n>m$, there are only finitely many $(n,m)$ such 
that all primes dividing $n m (n-m)$ are within $S$.    The recent 
technique of Chebyshev covers \cite{Ro2} 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 $S = \{\infty,2,3\}$.  

 \paragraph*{The finiteness conjecture.}       Based on the considerations just presented,
 we make the following conjecture.  
\begin{conjecture}  
\label{mainconj}
Let $F$ be a number field and let $S$ be a finite set of places of $F$.  
Then the set $\mbox{{\rm $\Fields$}}^{\rm big}_{F,S}$ is finite.  
\end{conjecture}
\noindent  In other words, while $\phi_{F,S}^{\rm small}$ is usually
infinite, we expect $\phi_{F,S}^{\rm big}$ to always be finite.  

 \paragraph*{Heuristic support for the finiteness conjecture.} 
 Bhargava \cite{Bh} 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 $|d| \rightarrow \infty$ 
 agrees with the previously known  Davenport-Heilbronn theorem in $n=3$. 
  In fundamental work,  Bhargava has
 proven the analogous theorem for $n=4$ \cite{Bh4} 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''
\begin{equation}
\label{Bhargava}
\phi^{\rm big}_{F,n,S} \approx \frac{1}{2}  \prod_{v \in S} \lambda_{F_v,n},
\end{equation}
where here we require that all Archimedean places are in $S$.   
Both $\lambda_{\R,n}$ and $\lambda_{\C,n}$ decrease superexponentially with $n$,
 according 
to \eqref{ArchAsymp2}.     For each ultrametric place
$v$ of $F$, the sequence $\lambda_{F_v,n}$ increases only exponentially, with growth factor 
$Q_v^{1/(p_v-1)}$.    All together, $\prod_{v \in S} \lambda_{F_v,n}$
decreases 
superexponentially with $n$.  This is much more than the mere convergence of 
$\sum_n  \prod_{v \in S}  \lambda_{F_v,n}$ 
which would be enough to heuristically support Conjecture~\ref{mainconj}.  
Note that one has a heuristic product formula \eqref{Bhargava} 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.  


 
 %    Bhargava's heuristic requires separating by Galois groups.   
%       In the quartic case, note that $S_4$ and $D_4$ both
%have five conjugacy classes.  The product heuristic \eqref{Bhargava}
%is appropriate if one considers $S_4$ and $A_4$ fields together as
%we are doing.   One has another product heuristic for 
%$D_4$, $C_4$, and $V$ together.   In our setting, suppose all archimedean
%places and also all places of residual characteristic $2$ and $3$ are in  $S$.
%Then one expects
%\begin{align*}
%\phi^{\rm big}_{F,4,S} &  \approx c_F^{\rm big} 5^{|S|}, &
%\phi^{\rm small}_{F,4,S} &  \approx c_F^{\rm small} 5^{|S|},
%\end{align*}
%with e.g.\, $c_\Q^{\rm big} = 33/100$ and $c_\Q^{\rm small} = 93/200$.
%Bhargava's setting of sorting by absolute discriminant is 
 


     We understand the factor $1/2$ in 
\eqref{Bhargava} in two different ways, depending on whether $n=2$ or $n \geq 3$. 
The case $n=2$ is best first explained in the simplified setting 
$F = \Q$ and $\{\infty,2\} \subseteq S$.  Then $\lambda_{\Q_v,2}$ is $1$, $4$, or $2$ according
to whether $v$ is  $\infty$, $2$, 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 
\eqref{Bhargava} 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 $n \geq 3$, one uses the local signs  $(2,d_v) HW(K_v)$ 
associated to 
$K/F \in \Algebras_{F,n,S}$ and a place $v \in S$.    
While these signs are all $1$ in the case $n=2$, in general
they can be $1$ or $-1$.    For $n \geq 3$, 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 \cite[Section~3.3]{JR}.  

\paragraph*{Comparison with computational results over $\Q$.}   Figure~\ref{compare} 
summarizes known facts about the case
 $(F,S) = (\Q,\{\infty,2,3\})$.  For $n = 1$, 2, 3, 4, 5, 6,  and 7, Jones and Roberts \cite{JR6}, \cite{JR7} evaluated  
 $\phi^{\rm big}_{\Q,\{2,3,\infty\}}$ to $1$, $3.5$, $8.\overline{3}$, $22$, $5$, $54$, 
 and $10$.    Roberts \cite{Ro} 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 \cite{MR} found 300 more fields in degree $9 \leq n \leq 33$ and 
 discussed the issue of finiteness of $\phi^{\rm big}_{\Q,S}$  
noncommittally as an open question.    Roberts \cite{Ro2} found $43$ more fields in
degrees $12 \leq n \leq 64$.  
\begin{figure}[htb]
\begin{center}
\epsfig{file=compare.eps,width=6in}
%\includegraphics[width=6in]{compare}
\parbox{5in}{\caption{\label{compare} Evaluations (black) and lower bounds (gray) for 
 $\phi^{\rm big}_{\Q,n,\{\infty,2,3\}}$,
compared with $\frac{1}{2} \lambda_{\R,n} \lambda_{2,n} \lambda_{3,n}$, with logarithmic vertical scale.
}}
\end{center}
\end{figure}

\paragraph*{Low discriminant phenomena and exceptional fields.}
Figure~\ref{compare} also compares the above computational results  
 with the more theoretical quantity 
 $\frac{1}{2} \lambda_{\R,n} \lambda_{2,n} \lambda_{3,n}$.   
 Although we are confident in Conjecture~\ref{mainconj} on
 a qualitative level, the situation remains enigmatic on a  quantitative
 level.   
 
      We interpret the poor agreement in degrees $\leq 7$ 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 \cite{Cubics} 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 \cite{Ro2} is very special in nature. 
 Define a field in $\Fields^{\rm big}_{F,n,S}$ to be 
{\em exceptional} if $\phi^{\rm big}_{F,n',S}<1$ for all
 $n' \geq n$.   The starting point $N(F,S)$ of the exceptional range
 is not as artificial as may first seem, because the decay of 
 $\phi^{\rm big}_{F,n,S}$ is rapid once it begins.  
 
    For $F = \Q$ and $S = \{\infty,2,3\}$, $\{\infty,3,5\}$, and $\{\infty,2,5\}$ 
    the exceptional
range starts at $N(\Q,S) = 62$, $38$, and $49$ respectively.    The
field constructed in \cite{Ro2} 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
 $\pm 2^a 5^b$ are exceptional if, as strongly expected, their Galois groups 
 are the full symmetric group on the degree. 

\paragraph*{Comparison with computational results over quadratic fields.}   In general, 
let $S$ be a set of rational places containing $\infty$.  Let $F$ be any degree $n_0$ 
number field.  
 For $v$ a place of $\Q$, 
let $\lambda_{F_v,n} = \prod \lambda_{F_w,n}$, the product being over
places $w$ mapping to $v$.   Then $\lambda_{F_\infty,n}^{1/n} \sim (e/n)^{n_0}$,
independently of the splitting behavior of $\infty$ in $F$.  Similarly, 
$\lambda_{F_p,n}^{1/n} \sim p^{n_0 /(p-1)}$ independently of the splitting
behavior of $p$.    However, looking at the subexponential factors suggests that
$\lambda_{F_v,n}$ 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 $S_d \subset \cS(F_d)$ be the set
of places mapping to $\infty$, $2$, or $3$ in $\cS(\Q)$.    So $|S_d| = 3$ for 
$d \in \{-6,-3,-1\}$ as none of $\infty$, $2$, or $3$ is split in $F_d$.  In contrast,
$|S_d|=4$ if $d \in \{-2,2,3,6\}$ as $3$ is split in $F_{-2}$ and $\infty$ is
split in the remaining cases.  

\begin{table}[htb]
\begin{center}
\parbox{5in}{\caption{\label{relativetable} Evaluations and lower bounds for 
$\phi^{\rm big}_{\Q(\sqrt{d}),n,\{\infty,2,3\}_d}$ in italics
 under $d$ compared
with $\frac{1}{2} \lambda_{\Q(\sqrt{d})_\infty,n} \lambda_{\Q(\sqrt{d})_2,n} \lambda_{\Q(\sqrt{d})_3,n}$ in plain type under $\prod$.} }
\end{center}
{\renewcommand{\arraycolsep}{3pt}
\[
\begin{array}{|r|rr|rr|rr|rrrr|rr|rrrr|}
\cline{8-17}
\multicolumn{7}{c|}{\;} & \multicolumn{4}{c|}{\mbox{No $v$ split}} &
\multicolumn{2}{c|}{\mbox{$3$ split}} &
\multicolumn{4}{c|}{\mbox{$\infty$ split}} \\
\hline
n & \C &  \R \cdot \R &  4 & 2 \cdot 2 & 9 & 3 \cdot 3 & -6 & -3 & -1 & \prod & -2 & \prod & 2 & 3 & 6 & \prod  \\ 
\hline
2 &  0.500 & 1.000  & 8 & 16 & 2 & 4  & \mathit{3.5} & \mathit{3.5} & \mathit{3.5} &4 & \mathit{7.5} &8& 
\mathit{7.5} & \mathit{7.5} & \mathit{7.5} & 8 \\
3 & 0.167 & 0.444 & 9 & 25 & 27 & 81  & \mathit{14.\overline{3}}  & \mathit{14.\overline{3}}   &  \mathit{14.\overline{3}} & 20 & \mathit{46.\overline{3}}  & 61 & \mathit{40.\overline{3}} & \mathit{40.\overline{3}} & \mathit{40.\overline{3}} & 54\\ 
4 &  0.042 & 0.174 & 272 &1296 & 29 & 121 & \mathit{87} & \mathit{87} &\mathit{87} &164 &
 \mathit{343} &686 & \mathit{385} & \mathit{385} & \mathit{385} & 685  \\
5 & 0.008 &  0.047  & 280 & 1600 & 55 & 361 &  & \mathit{17} & \mathit{21} &
  64 &&421& \mathit{\geq 87} &  &  & 361  \\
\hline
\end{array}
\]
}
\end{table}


    Table~\ref{relativetable} compares the two sides of \eqref{Bhargava} 
    in degrees  $2 \leq n \leq 5$.  The first pair of columns gives 
 $\lambda_{\C,n} < \lambda_{\R,n}^2$ to three decimal places.   
 The next two pairs of columns likewise give $\lambda_{4,n} < \lambda_{2,n}^2$
 and $\lambda_{9,n} < \lambda_{3,n}^2$.  The column 
 $\lambda_{2,n}^2$ 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 $\phi^{\rm big}_{\Q(\sqrt{d}),n,\{\infty,2,3\}_d}$ in italics,
 with the corresponding product 
 $\frac{1}{2} \lambda_{\Q(\sqrt{d})_\infty,n} \lambda_{\Q(\sqrt{d})_2,n} \lambda_{\Q(\sqrt{d})_3,n}$
 rounded to the nearest integer in regular type to its right.   The italicized masses 
 are computed from  \cite{JR6} for $n=2,3$ fields and from  \cite{Dr} 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:
 \begin{equation*}
 \begin{array}{r | l | l}
 2n & \mbox{One field} & \mbox{Two conjugate fields} \\
 \hline 
 4 & C_4, V & D_4 \\
 6 &   A_3C_2^*, \; S_3^tC_2^*, \; S_3C_2 & T5^*, \; T9,\; T10,\; T13 \\
 8 &   A_4C_2^*, \; S_4^tC_2^*, \;  S_4C_2& T33^*,\; T34,\;   T41, \;  T42^*,\;  T45, \; T46, \; T47\\
 10 & A_5C_2^*, \; S_5^tC_2^*,  \; S_5C_2& T40^*,\; T41,\;  T42,\; T43 \\
 \end{array}
 \end{equation*}
 Here the last group on each list is always $S_n^2.C_2$.  The starred groups
are the ones giving rise to $\Gal(K^g/\Q(\sqrt{d})) \cong A_n$.   This distinction
plays a role in the construction of Table~\ref{relativetable} 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 $\mathit{17}$, $\mathit{23}$, $\mathit{17}$, 
 $\mathit{49}$, $\mathit{41}$, $\mathit{41}$, and $\mathit{43}$.
 
Table~\ref{relativetable} reflects again how \eqref{Bhargava} does not take into account
low discriminant phenomena and consequently $\frac{1}{2} \prod_{v \in S} \lambda_{\Q(\sqrt{d})_v,n}$ is
an overestimate for the total mass $\phi^{\rm big}_{F_d,n,S_d}$.   
However, Table~\ref{relativetable} 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.  
  
 

 
 


\section{Contrast with positive characteristic}
\label{Contrast}
         It is standard in number theory to talk about global fields, meaning either number fields as in Section~\ref{NumberFields} 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~\ref{RC}-\ref{adic2}.
 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.  

\paragraph*{Allowing $e_0=\infty$.}         To accommodate positive characteristic, Sections~\ref{Wild} and \ref{Three} can be extended 
 by allowing $e_0 = \infty$ also.  Ore numbers in this situation
 are simpler.  Namely, as before,  $\Ore(p,\infty,e)$ 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  $\Ore(p,\infty,e)$ 
  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 
 $d(p,\infty,e,s) = \lceil s/p \rceil$.  Since all Ore numbers $s$ are non-maximal, 
 $W(p,\infty,e,s)$ always consists of the vectors in $\overline{\F}_p^{d(p,\infty,e,s)}$ with
 first component nonzero.  The function $\Lambda_{p,\infty,f_0}(x,y,z)$ is defined 
 as before.     Writing $s = s_0 p + s_1$, with $1 \leq s_1 \leq p-1$, it takes the explicit form
 \begin{equation}
 \label{charpLambda}
 \Lambda_{p,\infty,f_0}(x,y,z) = 
 \prod_{s_0=0}^\infty \prod_{s_1 = 1}^{p-1}
 \frac{1 - q^{s_0} x^e y^{e-1} z^{p s_0 + s_1}}{1 - q^{s_0+1} x^e y^{e-1} z^{p s_0 + s_1}},
 \end{equation}
 with $q = p^{f_0}$, as always.  
 
 \paragraph*{A contrast between the two specializations.}         The two situations in Section~\ref{One} now present a sharp contrast.  The Serre mass formula continues to hold: $\Lambda_{p,\infty,f_0}(x,1,1/q) = \Lambda(x)$, with essentially the same proof.
  Our formula $\Lambda_{p,\infty,f_0}(x,1,1) = \Lambda_Q(x)$ becomes completely degenerate as
 $Q = p^{e_0f_0}$ needs to be regarded as $\infty$.  The
coefficient of $x^n$ on both sides is $\lambda_{n}$ if $n<p$ and $\infty$ otherwise.  
There is no reasonable analog of Section~\ref{Core} or \ref{Asymptotics}.  

\paragraph*{Similar local behavior.}  
          Section~\ref{RC}-\ref{adic2} go through in the positive characteristic setting so that
 in particular the field-theoretic quantity $\Lambda_F(x,y,z)$ agrees with the directly
 defined  $\Lambda_{p,\infty,f_0}(x,y,z)$ of \eqref{charpLambda}.   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 $u = \sum_{i=1}^\infty b_i v^i$.  The smallest $i$ such that $b_i \neq 0$ is 
 just the degree $n=e$.     The smallest $i$ not divisible by $p$ such that $b_i \neq 0$
 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 $i b_i v^{i-1}$ is non-zero.   
 
 
  
 \paragraph*{Ease of constructing of   fields in $\mbox{\rm $\Fields$}^{\rm big}_{F,n,S}$.}
          The statement of Conjecture~\ref{mainconj} 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 
\begin{equation}
\label{genform}
g(x) =  a(x)^p + x + b(t)
\end{equation}
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 \eqref{discform} 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 $F_\infty = \F_p((1/t))$ of $\F_p(t)$.    If the algebra 
$F_\infty[x]/g(x)$ 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~\ref{mainconj}, consider the very
  special case $g_k(x) = x^2+x+t^{2k+1}$ for $k$ a non-negative integer.
 Changing variables via $x = 1/(y s^k)$ 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 $c = c_t + c_w = (1) + (2 k +1) = 2k+2$.  
 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 
 $\Fields_{F,2,\{\infty\}}^{\rm big}$.    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 \eqref{genform} to extensions 
 of $\F_p(t)$ have 
 Galois group all of $A_n$ or $S_n$.  
 
           Another simple specialization of \eqref{genform} is $g_{p,k}(x) = x^{k p} - x + t$ 
 for $p$ a prime and $k$ a positive integer.     Changing variables via $x = 1/y$ and
 $t = 1/s$, the equation becomes $y^{k p} - s y^{k p -1} + s$.   This is an Eisenstein 
 polynomial with conductor $c = c_t + c_w = (kp-1) + (kp-1) = 2 k p - 2$.    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 $t = - x^{k p} - x$.  
 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 \eqref{Jordan}
  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
  \cite[Theorems~6.6 and 6.3]{Ab}.  So the evidence is strong that for 
  each $p$, the field $\F_p(t)[x]/g_{p,k}(x)$ is in $\Fields^{\rm big}_{\F_p(t),pk,\{\infty\}}$ 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~\ref{mainconj} in characteristic
 zero gives two reasons why the corresponding statement fails in positive
 characteristic.  Again one can interpret $\frac{1}{2} \prod_{v \in S} \lambda_{F_v,n}$ 
 as the expected total mass of fields in $\Fields^{\rm big}_{F,n,S}$.  But in the 
 function field case if $n \geq p$ then each of the factors 
 $\lambda_{F_v,n}$ 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 $\lambda_{F_v,n}$ is replaced by a finite number 
 $\lambda^*_{F_v,n}$, the numbers $\frac{1}{2} \prod_{v \in S} \lambda^*_{F_v,n}$
 can still increase due to the lack of Archimedean places.   This is the
 phenomenon behind the existence of the family of polynomials $g_{p,k}(x)$.  




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\end{thebibliography}



\bigskip
\hrule
\bigskip

\noindent 2000 {\it Mathematics Subject Classification}: Primary 11S15;
Secondary 11P72, 11R21.\\

\noindent {\it Keywords}:  wild, partition, $p$-adic, ramified, mass.

\bigskip
\hrule
\bigskip


\noindent (Concerned with sequences \seqnum{A000041}, \seqnum{A000085}, \seqnum{A010054},\seqnum{A033687}, \seqnum{A131139}, and \seqnum{A131140}.)


\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received March 28 2007;
revised version received June 18 2007.
Published in {\it Journal of Integer Sequences}, June 18 2007.

\bigskip
\hrule
\bigskip

\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}.
\vskip .1in

\end{document}



 
