Inventiones mathematicae

, Volume 157, Issue 2, pp 275-328

First online:

Geometric deformations of modular Galois representations

  • Mark KisinAffiliated withDepartment of Mathematics, University of Chicago

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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.