Jos Brakenhoff (Leiden): Counting subrings of maximal orders

For a field extension K of the rational field Q of finite degree, the integral closure of Z in K is the largest finite extension of Z in K. We call this ring the maximal order of K and denote it by O_K.

For such a maximal order we define f_K(m) to be the number of subrings R of O_K for which the index [O_K:R] is m. If we fix the degree n of K over Q, then the maximum f(n,m) over all f_K(m) is finite.

In this talk we will study the limit behaviour of f(n,m) for fixed n when m goes to infinity.