On Gouvêa's conjecture in the unobstructed case
Fix an odd prime number \(p\). Let \(\mathbb{K}\) be a finite field of characteristic \(p\), and \(N\) a positive integer prime to \(p\). Let \(S= \{\text{the prime divisors of }N_p\}\cup\{\infty\}\) and \(G_S\) be the Galois group of the maximal Galois extension of \(\mathbb{Q}\) unramified outside \(S\). Let \(\overline\rho: G_S\to GL_2(\mathbb{K})\) be an absolutely irreducible residual representation which is associated to a classical eigenform of tame level \(N\) and weight \(k\) with any slope defined over the ring of integers \(O\) of a totally ramified finite extension \(K\) over the fraction field of the Witt ring \(W(\mathbb{K})\). B. Mazur showed that there exists a complete Noetherian local ring \(R(\overline\rho, S)\) with residue field \(\mathbb{K}\) and a universal deformation \(\rho^{\text{univ}}\) of \(\overline\rho: \rho^{\text{univ}}: G_S\to GL_2(R(\overline\rho, S))\). Let \(\rho^{\text{mod}}: G_S\to GL_2(\mathbb{T}(\overline\rho, N))\) denote the universal deformation of \(\overline\rho\) associated to Katz's \(p\)-adic modular forms. In particular, there exists a natural surjective homomorphism \(R(\overline\rho, S)\to\mathbb{T}(\overline\rho, N)\), and Gouvêa made the conjecture: \(R(\overline\rho, S)\simeq \mathbb{T}(\overline\rho, N)\). The author proves Gouvêa's conjecture in the unobstructed case (Theorem 1). For the proof he generalizes the theory of infinite ferns over \(\mathbb{Q}_p\) to the case that the ``universal deformation space'' is defined over \(K\) (Section 3). Note that \textit{G. Böckle} [Am. J. Math. 123, 985-1007 (2001; Zbl 0984.11025)] also showed Gouvêa's conjecture by other methods.
scientific article; zbMATH DE number 1902895