Abstract
Let f be a newform on Γ1(N), and V f the 2-dimensional p-adic Galois representation attached to f. Let S be a finite set of primes containing the primes divisors of Np, and denote by adV f the adjoint of V f . Under some mild conditions on f, we show that H1 g (Gℚ,S,adV f )=0.
Using this result, we show that the universal deformation space of the residual representation attached to f is smooth and 3-dimensional at the point corresponding to f. When f has finite slope, one can also use this result to give a deformation theoretic description of the “eigencurve” of Coleman-Mazur at the point corresponding to f.
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Kisin, M. Geometric deformations of modular Galois representations. Invent. math. 157, 275–328 (2004). https://doi.org/10.1007/s00222-003-0351-2
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DOI: https://doi.org/10.1007/s00222-003-0351-2