Counting two-step nilpotent wildly ramified extensions of function fields
Fabian Gundlach, Béranger Seguin
TL;DR
This work develops a local–global counting framework for wildly ramified $p$-group extensions of function fields in characteristic $p>2$, counting by the last ramification jump rather than discriminants. By harnessing Abrashkin’s nilpotent Artin–Schreier theory and a Lazard/Lie-algebra parametrization via Witt vectors, the authors build a local counting theory with fundamental domains and ramification equations, then glue the local data into global asymptotics using a precise analytic lemma. The main contribution is an exact local–global principle for the last jump and an explicit global asymptotic formula, including detailed results for Heisenberg-type groups and a broader class of exponent-$p$ groups. The findings illuminate how wild ramification data controls the distribution of Galois groups in function fields and establish a robust method to translate local ramification information into global counts, with potential further refinements via geometric and moduli-space perspectives.
Abstract
We study the asymptotic distribution of wildly ramified extensions of function fields in characteristic $p > 2$, focusing on (certain) $p$-groups of nilpotency class at most $2$. Rather than the discriminant, we count extensions according to an invariant describing the last jump in the ramification filtration at each place. We prove a local-global principle relating the distribution of extensions over global function fields to their distribution over local fields, leading to an asymptotic formula for the number of extensions with a given global last-jump invariant. A key ingredient is Abrashkin's nilpotent Artin-Schreier theory, which lets us parametrize extensions and obtain bounds on the ramification of local extensions by estimating the number of solutions to certain polynomial equations over finite fields.