Lame curves with bad reduction

Abstract

Lame curves are a particular class of elliptic curves (with a torsion point attached to them) which naturally arise when studying Lame operators with finite monodromy. They can be realized as covers of the projective line unramified outside three points and can be defined over number fields. This paper investigates their p-adic properties. The main ingredient is formal/rigid geometry and in particular the use of p-adic theta functions. As a consequence, we can completely enumerate Lam\'e curves with bad reduction (by giving, among the others, the p-adic valuation of their j-invariant) and describe the (local) Galois action.