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Density of potentially crystalline representations of fixed weight
- Part of
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- Journal:
- Compositio Mathematica / Volume 152 / Issue 8 / August 2016
- Published online by Cambridge University Press:
- 23 May 2016, pp. 1609-1647
- Print publication:
- August 2016
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- Article
- Export citation
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Let
$K$ be a finite extension of 
$\mathbb{Q}_{p}$ and let 
$\bar{\unicode[STIX]{x1D70C}}$ be a continuous, absolutely irreducible representation of its absolute Galois group with values in a finite field of characteristic 
$p$. We prove that the Galois representations that become crystalline of a fixed regular weight after an abelian extension are Zariski-dense in the generic fiber of the universal deformation ring of 
$\bar{\unicode[STIX]{x1D70C}}$. In fact we deduce this from a similar density result for the space of trianguline representations. This uses an embedding of eigenvarieties for unitary groups into the spaces of trianguline representations as well as the corresponding density claim for eigenvarieties as a global input.
