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Zbl 1176.11055
Mézard, Ariane
Obstructions to deformations of reducible Galois representations and class groups. (Obstructions aux déformations de représentations galoisiennes réductibles et groupes de classes.) (French)
[J] J. Théor. Nombres Bordx. 17, No. 2, 607-618 (2005). ISSN 1246-7405

Summary: In this paper, we develop a new strategy to understand the obstructions to deformations of reducible odd 2-dimensional global Galois representations $\overline\rho$. It is known that these obstructions are localized in a Shafarevich group. After [{\it G. Böckle} and {\it A. Mézard}, J. Number Theory 78, No. 2, 167--203 (1999; Zbl 0958.11040)] these obstructions are related with several classical conjectures (Vandiver's conjecture, Greenberg's conjecture). The idea of this note is to introduce another Shafarevich group depending on the field $L$ fixed by $\text{ker}\,\overline\rho$. We then compare the two groups by taking the co-invariant by $\text{Im}\,\overline\rho$. This strategy yields new conditions for the vanishing of the obstructions in terms of class groups of $L$.

MSC 2000:: *11R32 Galois theory for global fields
11F80 Galois properties

Citations: Zbl 0958.11040

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