Lifting of nearly extraordinary Galois representations

Constructing certain special analytic Galois extensions

“…Observe that since G p acts diagonally, the 1 dimensional space Diag(ρ ↾Gp ) is a summand of Ad 0ρ ↾Gp and Ad 0ρ Proof. That F p is liftable follows from Lemma 4.5 of [1]. We show that it is balanced, i.e.…”

“…where ϕ 1 and ϕ 2 are two characters (possibly ramified) 1) . The tangent space N p is defined as the set of deformations of ρ to the dual numbers F p (F p [ǫ]) which has a natural structure of an F p vector space and is realized as a subspace of H 1 (G p , Ad 0 ρ).…”

“…• 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.…”