On deformation rings and Hecke rings.

*(English)*Zbl 0867.11032The author proves the following result: Let \(E\) be an elliptic curve defined over the rational field \(\mathbb{Q}\) with semistable reduction at the prime 3. Suppose that either \(E\) has semistable reduction at 5 or that \(\rho_{E,3}: G_{\mathbb{Q}}\to\text{GL}(2,\mathbb{F}_3)\) is absolutely irreducible when restricted to \(\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}(\sqrt{-3}))\). Then \(E\) is modular.

In his fundamental paper [Ann. Math., II. Ser. 141, 443-551 (1995; Zbl 0823.11029)], A. Wiles formulates a conjecture identifying certain Hecke rings as universal deformation rings for modular Galois representations \(\rho:\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to \text{GL}_2(\overline{\mathbb{F}}_\ell)\), where \(\ell\) is an odd prime. Under some hypotheses for \(\rho\), Wiles proves that the minimal deformation ring and Hecke ring are isomorphic. The proof uses some of his results with Taylor.

In the paper under review, the author extends the result of Wiles. He removes those hypotheses of Wiles on the local behaviour of \(\rho\) at the primes \(p\neq\ell\).

The work of K. Ribet [Invent. Math. 100, 431-476 (1990; Zbl 0773.11039)] on the refined conjecture of Serre, as well as an adaptation of Ribet’s proof by the author [Elliptic curves, modular forms, and Fermat’s last theorem, Proc. Conf. Hong Kong 1993, 22-37 (1995; Zbl 0853.11031)], are used to attach to the representation \(\rho_0\) \((\rho_0\otimes\overline{\mathbb{F}}_\ell=\rho\)) a modular form in \(S_2(\Gamma_Q)\), where \(\Gamma_Q\) is a certain group intermediate to \(\Gamma_1(N_Q)\) and \(\Gamma_0(N_Q)\), and \(N_Q\) is an integer which raises the conductor \(N_0\) of \(\rho_0\).

Let \(P\) denote the set of primes \(p\equiv-1\text{ mod }\ell\) such that \(\rho_0\) restricted to the decomposition group \(D_p\) is irreducible and that \(\rho_0\) restricted to the inertia subgroup \(I_p\subseteq D_p\) reduces. In the paper of Wiles, it is assumed that \(P=\emptyset\). In the present case, it is assumed that \(P\neq\emptyset\). This forces to work with modular forms on quaternion algebras \(B\) ramified at exactly the primes in \(P\) (if \(\#P\) is even), or at \(P\cup\{\infty\}\) (if \(\#P\) is odd). Some of the ingredients needed for the analogue of the “minimal case” are supplied by appealing to the Jacquet-Langlands correspondence.

To prove the main results in the paper, the methods follow those of Wiles and Taylor. Faltings’ argument in Wiles paper is adapted to the present context. A complete proof that includes the necessary modifications is also given.

In his fundamental paper [Ann. Math., II. Ser. 141, 443-551 (1995; Zbl 0823.11029)], A. Wiles formulates a conjecture identifying certain Hecke rings as universal deformation rings for modular Galois representations \(\rho:\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to \text{GL}_2(\overline{\mathbb{F}}_\ell)\), where \(\ell\) is an odd prime. Under some hypotheses for \(\rho\), Wiles proves that the minimal deformation ring and Hecke ring are isomorphic. The proof uses some of his results with Taylor.

In the paper under review, the author extends the result of Wiles. He removes those hypotheses of Wiles on the local behaviour of \(\rho\) at the primes \(p\neq\ell\).

The work of K. Ribet [Invent. Math. 100, 431-476 (1990; Zbl 0773.11039)] on the refined conjecture of Serre, as well as an adaptation of Ribet’s proof by the author [Elliptic curves, modular forms, and Fermat’s last theorem, Proc. Conf. Hong Kong 1993, 22-37 (1995; Zbl 0853.11031)], are used to attach to the representation \(\rho_0\) \((\rho_0\otimes\overline{\mathbb{F}}_\ell=\rho\)) a modular form in \(S_2(\Gamma_Q)\), where \(\Gamma_Q\) is a certain group intermediate to \(\Gamma_1(N_Q)\) and \(\Gamma_0(N_Q)\), and \(N_Q\) is an integer which raises the conductor \(N_0\) of \(\rho_0\).

Let \(P\) denote the set of primes \(p\equiv-1\text{ mod }\ell\) such that \(\rho_0\) restricted to the decomposition group \(D_p\) is irreducible and that \(\rho_0\) restricted to the inertia subgroup \(I_p\subseteq D_p\) reduces. In the paper of Wiles, it is assumed that \(P=\emptyset\). In the present case, it is assumed that \(P\neq\emptyset\). This forces to work with modular forms on quaternion algebras \(B\) ramified at exactly the primes in \(P\) (if \(\#P\) is even), or at \(P\cup\{\infty\}\) (if \(\#P\) is odd). Some of the ingredients needed for the analogue of the “minimal case” are supplied by appealing to the Jacquet-Langlands correspondence.

To prove the main results in the paper, the methods follow those of Wiles and Taylor. Faltings’ argument in Wiles paper is adapted to the present context. A complete proof that includes the necessary modifications is also given.

Reviewer: P.Bayer (Barcelona)