Modularity of a family of elliptic curves
The recent work of \textit{A. Wiles} [Ann. Math., II. Ser. 141, No. 3, 443-551 (1995; Zbl 0823.11029)] and \textit{R. Taylor} and \textit{A. Wiles} [ibid. 553-572 (1995; Zbl 0823.11030)] shows that every elliptic curve over \(\mathbb{Q}\) with semistable reduction at every prime is modular. Building on this work, \textit{F. Diamond} [Ann. Math., II. Ser. 144, No. 1, 137-166 (1996; Zbl 0867.11032)] showed that in fact semistability at 3 and 5 implies modularity. Rubin and Silverberg noted that every elliptic curve over \(\mathbb{Q}\) with all of its 2-torsion points defined over \(\mathbb{Q}\) -- that is, every elliptic curve with a model of the form \(y^2=(x-a)(x-b)(x-c)\) for some rational numbers \(a\), \(b\), and \(c\) -- has a twist that is semistable outside 2, and is therefore modular by the first author's result. The authors of the present paper show that the modularity of an elliptic curve with all of its 2-torsion defined over \(\mathbb{Q}\) can be proven directly from the work of Wiles and Taylor, without reference to the first author's refinement. Suppose \(E\) is an elliptic curve over \(\mathbb{Q}\) with all of its 2-torsion points defined over \(\mathbb{Q}\). As per Rubin and Silverberg, the authors replace \(E\) with a twist \(E'\) that has semistable reduction outside 2. If \(E'\) has semistable reduction at 2 as well, then \(E'\) has semistable reduction everywhere, and Wiles's theorem 0.4 [op. cit.] shows that \(E'\) is modular. The authors show that if \(E'\) has additive reduction at 2, then for every odd prime \(\ell\) the \(\ell\)-adic Galois representation attached to \(E'\), restricted to an inertia group at 2, is absolutely irreducible. From this they deduce that the hypotheses of Wiles's theorem 0.3 are satisfied, and once again the conclusion is that \(E'\) is modular. In their arguments, the authors spell out some details of the proof of Wiles's theorem 0.3 that Wiles himself leaves to the reader.
scientific article; zbMATH DE number 817755