Demuskin groups with group actions and applications to deformations of Galois representations

Authors
Citation
G. Bockle, Demuskin groups with group actions and applications to deformations of Galois representations, COMP MATH, 121(2), 2000, pp. 109-154
Citations number
28
Categorie Soggetti
Mathematics
Journal title
COMPOSITIO MATHEMATICA
ISSN journal
0010437X → ACNP
Volume
121
Issue
2
Year of publication
2000
Pages
109 - 154
Database
ISI
SICI code
0010-437X(200004)121:2<109:DGWGAA>2.0.ZU;2-U
Abstract
We determine the universal deformation ring, in the sense of Mazur, of a re sidual representation <(rho)over bar> :G(K) --> GL(2)(k), where k is a fini te field of characteristic p and K is a local field of residue characterist ic p. As one might hope for, but is not proven in the global case, the defo rmation ring is a complete intersection, flat over W(k), with the exact num ber of equations given by the dimension of H-2(G(K), ad(<(rho)over bar>)). We then go on to determine the ordinary locus inside the deformation space and, using ideas of Mazur, apply this to compare the universal and the univ ersal ordinary deformation spaces. Provided that the universal ring for ord inary deformations with fixed determinant is finite flat over W(k), as was shown in many cases by Diamond, Fujiwara, Taylor-Wiles and Wiles, we show t hat the corresponding universal deformation ring - with no restriction of o rdinariness or fixed determinant - is a complete intersection, finite flat over W(k) of the dimension conjectured by Mazur, provided that the restrict ion of det(<(rho)over bar>) to the inertia subgroup is different from the i nverse cyclotomic character.