### An "infinite fern" in the universal deformation space of Galois representations

#### Resum

I hope this article will be helpful to people who might want a quick overview of how modular representations fit into the theory of deformations of Galois representations.\newline There is also a more specific aim: to sketch a construction of a “point-set topological” configuration (the image of an “infinite fern”) which emerges from consideration of modular representations in the universal deformation space of all Galois representations. This is a configuration hinted at in [20], but now, thanks to some recent important work of Coleman [6, 7], it is something one can actually produce! The “infinite fern” is joint work with F.Q. Gouvêa, and will be the subject of slightly more systematic study in a future paper in which some consequences of its existence will be discussed.\newline Although the “infinite fern” which appears in the last section of these notes is hardly as profound a point-set topological object as some of the classic constructions of R.H. Bing, I would like to think that he might have nevertheless enjoyed it. I want to dedicate this article to him, in appreciation of his mathematics and of his energetic enthusiasm.