The small Schottky--Jung locus in positive characteristics different from two

We prove that the locus of Jacobians is an irreducible component of the small Schottky locus in any characteristic different from $2$. The proof follows an idea of B. van Geemen in characteristic $0$ and relies on a detailed analysis at the boundary of the $q$- expansion of the Schottky-Jung relations. We obtain algebraically such relations using Mumford's theory of $2$-adic theta functions. We show how the uniformization theory of semiabelian schemes, as developed by D. Mumford, C.-L. Chai and G. Faltings, allows the study of higher dimensional $q$-expansions simplifying the argument.