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.