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Journal Article

On automorphic points in polarized deformation rings


Allen,  Patrick B.
Max Planck Institute for Mathematics, Max Planck Society;

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Allen, P. B. (2019). On automorphic points in polarized deformation rings. American Journal of Mathematics, 141(1), 119-167. doi:10.1353/ajm.2019.0003.

Cite as: https://hdl.handle.net/21.11116/0000-0004-7693-1
For a fixed mod $p$ automorphic Galois representation, $p$-adic automorphic
Galois representations lifting it determine points in universal deformation space. In the case of modular forms and under some technical conditions, B\"{o}ckle showed that every component of deformation space contains a smooth modular point, which then implies their Zariski density when coupled with the infinite fern of Gouv\^{e}a-Mazur. We generalize B\"{o}ckle's result to the
context of polarized Galois representations for CM fields, and to two dimensional Galois representations for totally real fields. More specifically, under assumptions necessary to apply a small $R = \mathbb{T}$ theorem and an assumption on the local mod $p$ representation, we prove that every irreducible component of the universal polarized deformation space contains an automorphic point. When combined with work of Chenevier, this implies new results on the Zariski density of automorphic points in polarized deformation space in dimension three.