A local to global principle for densities over function fields

2017/01/04 by Micheli, Giacomo
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1701.01178

Abstract

Let d be a positive integer and \mathbb H be an integrally closed subring of a global function field F. The purpose of this paper is to provide a general sieve method to compute densities of subsets of \mathbb H^d defined by local conditions. The main advantage of the method relies on the fact that one can use results from measure theory to extract density results over \mathbb H^d. Using this method we are able to compute the density of the set of polynomials with coefficients in \mathbb H which give rise to "good" totally ramified extensions of the global function field F. As another application, we give a closed expression for the density of rectangular unimodular matrices with coefficients in \mathbb H in terms of the L-polynomial of the function field.

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