Neighborhoods of singular elliptic curves and the Riemann-Hilbert correspondence at infinity

Abstract

In his thesis work, the first author studies neighborhoods of singular elliptic curves of type $I_0^*$ (Kodaira's classification) and provides the analytic classification. This is related via a double covering construction to the analytic invariants found by Touzet, Voronin and the second author for the analytic classification of neighborhoods of smooth elliptic curves. After recalling these constructions, we explain in this note how this construction is conjecturally related to the Riemann-Hilbert correspondence at infinity for Painlev{\'e} VI case when considering neighborhood of Okamoto divisor.