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Unobstructed modular deformation problems. (English) Zbl 1071.11027
Let \(f=\sum a_nq^n\) be a newform of weight \(k>1\), level \(N\), character \(\omega\). Let \(S\) be a finite set of primes of \(\mathbb Q\) including infinity and the divisors of \(N\). Let \(K\) be the number field generated by the \(a_n\). For any prime \(\lambda\) of \(K\) let \(\overline\rho_{f,\lambda}\) be the Deligne-Serre representation in \(\text{GL}(2,k_\lambda)\) of the Galois group \(G_{\mathbb Q, S\cup\{\ell\}}\) of the maximal extension of \(\mathbb Q\) unramified outside \(S\) and the characteristic \(\ell\) of the residue field \(k_\lambda\) at \(\lambda\). This \(\overline\rho_{f,\lambda}\) is absolutely irreducible for almost all \(\lambda\). For such \(\lambda\) let \(R^S_{f,\lambda}\) denote the universal deformation ring parametrizing lifts of \(\overline\rho_{f,\lambda}\) to two dimensional representations over noetherian local rings with residue field \(k_\lambda\). This paper shows that if \(k>2\) then \(R^S_{f,\lambda}\) is \(W(k_\lambda)[[T_1,T_2,T_3]]\) for almost all \(\lambda\) in \(K\), where \(W(k)\) is the Witt ring of \(k\). Similar result holds for \(k=2\), and more precise results are obtained when the level is 1, trivial character and for certain weights. A. Yamagami also obtained some of these results.

11F80 Galois representations
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