On Gouvêa's conjecture on controlling the conductor
Fix an odd prime number \(p\). Let \(\overline{\rho}: G_{\mathbb{Q}}\to \text{GL}_2(\mathbb{F}_p)\) be an absolutely irreducible residual representation of conductor \(N= N(\overline{\rho})\). Let \(S:=\{\)the prime divisors of \(N_p\}\cup \{\infty\}\), and let \(G_S\) denote the maximal Galois extension of \(\mathbb{Q}\) unramified outside \(S\). Then \(\overline{\rho}\) factors through \(G_S\); i.e. \(\overline{\rho}: G_S\to \text{GL}_2 (\mathbb{F}_p)\). Mazur showed that there exists a complete noetherian local ring \(\mathbb{R}(\overline{\rho},S)\) with residue field \(\mathbb{F}_p\) and a universal deformation of \(\overline{\rho}\); namely, \(\rho^{\text{univ}}: G_S\to \text{GL}_2 (\mathbb{R}(\overline{\rho},S))\). On the other hand there exists a deformation of \(\overline{\rho}\) to the Hecke algebra \(\mathbb{T}(\overline{\rho},N)\) (Gouvêa). Thus we have a natural surjective homomorphism \(\mathbb{R}(\overline{\rho},S)\to \mathbb{T}(\overline{\rho},N)\). Gouvêa also defined the level \(N\) universal deformation ring \(\mathbb{R}(\overline{\rho},N)\), and showed that \(\mathbb{R}(\overline{\rho},S)\to \mathbb{T}(\overline{\rho},N)\) factors through \(\mathbb{R}(\overline{\rho},N)\). Conjecture [\textit{F. Gouvêa}, J. Number Theory 34, 95-113 (1990; Zbl 0705.11032)]. The surjective homomorphism \(\mathbb{R}(\overline{\rho},N)\to \mathbb{T}(\overline{\rho},N)\) is an isomorphism. The author proves Gouvêa's conjecture in a special case when \(\overline{\rho}\) is associated to a class of classical eigenforms of type \((N,k,1)\) \((k\geq 2)\) (Main Theorem). He actually shows that \(\mathbb{R}(\overline{\rho},S)\simeq \mathbb{T}(\overline{\rho},N)\) (Theorem 2.2). The main ingredients in the proofs are: the Lenstra-de Smit construction of universal deformation rings [\textit{B. de Smit} and \textit{H. W. Lenstra jun.}, in: Modular forms and Fermat's last theorem, G. Cornell et al. (eds.), 313-326 (1997; Zbl 0907.13010)] and a generalization of the main theorem of \textit{F. Gouvêa} and \textit{B. Mazur} [in: AMS/IP Stud. Adv. Math. 7, 127-142 (1998; Zbl 1134.11324)].
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