Deformations of locally abelian Galois representations and unramified extensions

Abstract. In a previous paper the authors showed that, under some technical conditions, the local Galois representations attached to the members of a non-CM family of ordinary cusp forms are indecomposable for all except possibly finitely many members of the family. In this paper we use deformation theoretic methods to give examples of non-CM families for which every classical member of weight at least two has a locally indecomposable Galois representation.

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“…We establish its existence in this section using ideas introduced by Mazur in [Maz89]. See also [Oht06].…”

Section: Universal Locally Split Deformation Ringmentioning

confidence: 98%

Locally Indecomposable Galois Representations

Ghate¹,

Vatsal

2011

Can. j. math.

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“…• L is unramified at primes above p and ramified above finitely many rational primes at which it is tamely ramified. Such p-adic extensions L were first constructed by Ohtani [6] and Blondeau [1] and their methods relied on lifting suitable irreducible Galois representations which are extraordinary at p. The construction in [6] and [1] relies on the existence of an eigenform f with companion forms (and thus extraordinary at p) such the image of the residual representation ρf contains SL 2 (F p ). Computations for p < 3500 show that there are precisely four primes 107, 139, 271 and 379 for which such an eigenform f exists.…”

Section: Introductionmentioning

confidence: 99%

Constructing certain special analytic Galois extensions

Ray

2020

Journal of Number Theory

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For every prime p ≥ 5 for which a certain condition on the class group Cl(Q(µ p )) is satisfied, we construct a p-adic analytic Galois extension of the infinite cyclotomic extension Q(µ p ∞ ) with some special ramification properties. In greater detail, this extension is unramified at primes above p and tamely ramified above finitely many rational primes and its Galois group over Q(µ p ∞ ) is isomorphic to a finite index subgroup of SL 2 (Z p ) which contains the principal congruence subgroup. For the primes 107, 139, 271 and 379 such extensions were first constructed by Ohtani and Blondeau. The strategy for producing these special extensions at an abundant number of primes is through lifting two-dimensional reducible Galois representations which are diagonal when restricted to p.

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Lifting of nearly extraordinary Galois representations

Blondeau

2013

Journal of Number Theory

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Deformations of locally abelian Galois representations and unramified extensions

Abstract: We study the deformation theory of Galois representations whose restriction to every decomposition subgroup is abelian. As an application, we construct unramified non-solvable extensions over the field obtained by adjoining all p-power roots of unity to the field of rational numbers.

Cited by 5 publications

References 19 publications

Locally Indecomposable Galois Representations

Locally Indecomposable Galois Representations

Constructing certain special analytic Galois extensions

Lifting of nearly extraordinary Galois representations

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