The refined conjecture of Serre. (English) Zbl 0853.11031

Coates, John (ed.) et al., Elliptic curves, modular forms, & Fermat’s last theorem. Proceedings of the conference on elliptic curves and modular forms held at the Chinese University of Hong Kong, December 18-21, 1993. Cambridge, MA: International Press. Ser. Number Theory. 1, 22-37 (1995).
Let \(\ell\) be a prime, and let \(F\) be an algebraic closure of \(F_\ell\). Suppose that \(\rho\colon \mathrm{Gal} (\overline{\mathbb Q}, \mathbb Q)\to \mathrm{GL}_2 (F)\) is a continuous irreducible representation with odd determinant. J.-P. Serre [Duke Math. J. 54, 179–230 (1987; Zbl 0641.10026)] conjectured that any such representation is obtained from a modular form, i.e., that for some weight \(k\) and level \(N\) there exists a modular eigenform of weight \(k\) on \(\Gamma_1 (N)\) whose associated Galois representation reduces to \(\rho\). In the same article, Serre formulates a refined version of this conjecture, in which the weight \(k= k(\rho)\) and the level \(N= N(\rho)\) are specified precisely. The author, following Ribet, calls this the “refined Serre conjecture”.
The goal of this paper is to prove, for \(\ell\ge 3\), that the refined Serre conjecture is equivalent to the (unrefined) Serre conjecture. In other words, the claim is that if there exists a modular form of some weight and level associated to \(\rho\), then there also exists a modular form of weight \(k(\rho)\) and level \(N(\rho)\). In fact, Serre also predicts the nebentypus character of the form, and (under a slight restriction when \(\ell=3\)) the author proves that one can also get the form associated to \(\rho\) to have the correct character. In his “Report on mod \(\ell\) representations” [in Motives, Proc. Symp. Pure Math. 55, Pt. 2, 639–676 (1994; Zbl 0822.11034)], K. Ribet put together his own work and the work of many other mathematicians to show that the proof of the equivalence of the two forms of the conjecture could be achieved if one could remove a restriction from a crucial intermediate result.
In this article, the author finishes the proof by removing the restriction in question. The article is closely based on Ribet’s article, and should be read in conjunction with it. In the final section, the author shows how a variant of these results that appears in the work of Wiles can be deduced from the main theorem.
For the entire collection see [Zbl 0824.00025].


11F11 Holomorphic modular forms of integral weight
11F80 Galois representations