Reconstruction of open subschemes of elliptic curves in positive characteristic by their geometric fundamental groups under some assumptions

Abstract

The goal of this paper is the reconstruction of the isomorphism class of a nonempty open subscheme of an elliptic curve over an algebraic closure of a finite field by its étale fundamental group under some assumptions. First we will prove a certain general property of elliptic curves over finite fields. This shows the existence of linear relations of the cusps. On the other hand, the author's previous work shows that the existence of linear relations of the cusps can be characterized by the étale fundamental group and a certain subgroup. By combining these results, we prove the main reconstruction result.