On the adjoint Selmer groups of semi-stable elliptic curves and Flach's zeta elements

Abstract

We explicitly construct the rank one primitive Stark (equivalently, Kolyvagin) system extending a constant multiple of Flach's zeta elements for semi-stable elliptic curves. As its arithmetic applications, we obtain the equivalence between a specific behavior of the Stark system and the minimal modularity lifting theorem, and we also discuss the cyclicity of the adjoint Selmer groups. This Stark system construction yields a more refined interpretation of the collection of Flach's zeta elements than the "geometric Euler system" approach due to Flach, Wiles, Mazur, and Weston.

Revised following the comments. The computation of the constant multiple is significantly simplified, and some errors are corrected. Comments are very welcome