Geometric deformations of modular Galois representations
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- Kisin, M. Invent. math. (2004) 157: 275. doi:10.1007/s00222-003-0351-2
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Let f be a newform on Γ1(N), and Vf 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 adVf the adjoint of Vf. Under some mild conditions on f, we show that H1g(Gℚ,S,adVf)=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.