Some computations with Hecke rings and deformation ring, with an appendix by Amod Agashe and William Stein

Abstract

In the proof by Wiles, completed by Taylor-Wiles, of the fact that all semistable elliptic curves over {\small $\QQ$} are modular, certain deformation rings play an important role. In this note, we explicitly compute these rings for the elliptic curve {\small $Y^2+XY=X^3-X^2-X-3$} of conductor 142.