## The 2-adic eigencurve is proper

### Summary

Coleman and Mazur ask whether the Eigencurve has any "holes". We answer their question in the negative for the $2$-adic Eigencurve of tame level one.

11F33, 11F85

### Keywords/Phrases

modular forms, modular curves, $U$-operator, overconvergent modular form