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**An “infinite fern” in the universal deformation space of Galois representations.**
*(English)*
Zbl 0865.11046

The aim of this article is twofold: to give a quick overview of how modular representations fit into the theory of deformations of Galois representations and 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. F. Q. Gouvêa and B. Mazur had already conjectured the existence of the “infinite fern” in [Math. Comput. 58, 793-806 (1992; Zbl 0773.11030)]. Now, thanks to some recent important work of R. Coleman [Invent. Math. 124, 215-241 (1996; Zbl 0851.11030); J. Théor. Nombres Bordx. 7, No. 1, 333-365 (1995); \(p\)-adic Banach spaces and families of modular forms, Preprint August (1995), Invent. Math. (to appear)], it is something one can actually produce!

A consequence of the existence of these infinite ferns (which will be written up in a future joint paper with Gouvêa) is that if \(\;\overline {\rho}: G_\mathbb{Q}\to GL_2 (\mathbb{F}_p)\) is an absolutely irreducible residual representation which comes from a modular form on \(\Gamma_0 (p)\), and for which the corresponding “unramified outside \(p\)” deformation problem is “unobstructed”, then the entire universal deformation ring attached to \(\overline {\rho}\) may be reconstructed from modular representations.

A consequence of the existence of these infinite ferns (which will be written up in a future joint paper with Gouvêa) is that if \(\;\overline {\rho}: G_\mathbb{Q}\to GL_2 (\mathbb{F}_p)\) is an absolutely irreducible residual representation which comes from a modular form on \(\Gamma_0 (p)\), and for which the corresponding “unramified outside \(p\)” deformation problem is “unobstructed”, then the entire universal deformation ring attached to \(\overline {\rho}\) may be reconstructed from modular representations.

Reviewer: B.Mazur (Cambridge / MA)

### MSC:

11F80 | Galois representations |

11F11 | Holomorphic modular forms of integral weight |

11F33 | Congruences for modular and \(p\)-adic modular forms |