Whittaker groups and hyperelliptic curves

TL;DR

This paper studies Whittaker groups and hyperelliptic curves over non-archimedean fields, providing explicit parametrizations, relations between fixed points and branch loci, and classifications of these groups with analytic reductions.

Contribution

It develops explicit theta function parametrizations of Whittaker curves and establishes a Galois covering relation between fixed points and branch points configurations.

Findings

Explicit theta function parametrization of Whittaker curves.

The morphism from fixed points to branch points is a Galois cover with group {b1}1^{d-1}.

Confirmed the main theoretical results through computations for genus 2 and 3 cases.

Abstract

Let K be a complete, non-archimedean valued field with a residue field of characteristic different from 2. A Whittaker group G is a discontinuous subgroup of PGL(2,K), freely generated by elements s_0,...,s_g of order two, each defined by a pair of fixed points {a_0,b_0},...,{a_g,b_g}. These fixed points are called ``in good position''. A subgroup W in G of index 2 is a Schottky group and produces a hyperelliptic Mumford curve Omega/W --> Omega/G = P^1, called `Whittaker curve', of genus g and with branch locus B in P^1(K). An explicit parametrization of Whittaker curves in terms of theta functions for W and G and the data of the fixed points, is developed. In particular, this allows one to express the branched points (and other data such as p-adic periods and p-adic heights) in terms of values of theta functions. A central theme of this paper is the relation between the fixed points and the branch locus. For a given configuration (P,m) of $g+1$ pairs of points in P^1, one defines a rigid space Fix_{P,m} of fixed points in good position with that configuration and a rigid space of branched points $ Branch_{P,m} in that configuration. A main result is that the natural morphism FB: Fix_{P,m} --> Branch_{P,m} is a rigid etale covering with Galois group {\pm 1}^{d-1} for some d>0. For all cases of genus g=2,3 (and for some more), an approximation of FB is computed which confirms the main result. Classification of Whittaker groups and analytic reductions of Whittaker curves is another important issue of this paper. The background material in this paper complements the work of L.~Gerritzen, G.~Van Steen, F.~Herrlich and others. It involves re-examination of some proofs, the derivation of properties of semi-stable analytic reductions and studying good position of fixed points.