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Report on mod \(\ell\) representations of \(\mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q)\). (English) Zbl 0822.11034

Jannsen, Uwe (ed.) et al., Motives. Proceedings of the summer research conference on motives, held at the University of Washington, Seattle, WA, USA, July 20-August 2, 1991. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 55, Pt. 2, 639-676 (1994).
Let \(\ell\) be an odd prime, and let \(\mathbb F\) be the algebraic closure of the finite field with \(\ell\) elements \(\mathbb F_\ell\). Let \(G= \mathrm{Gal}(\overline {\mathbb Q}/ \mathbb Q)\) be the absolute Galois group of \(\mathbb Q\), and let \(\rho\colon G\to \mathrm{GL}_2(\mathbb F)\) be a continuous irreducible representation. Suppose \(\rho\) is odd, that is, \(\det\rho(c) =- 1\), where \(c\) denotes a complex conjugation in \(G\). In this situation, Serre has conjectured that \(\rho\) arises from a modular form.
More precisely, the “Serre conjecture” is that there exists some level \(N\) and some weight \(k\) such that the representation \(p\) is attached to a new eigenform \(f\in S_k (\Gamma_1 (N))\). In [Duke Math. J. 54, 179–230 (1987; Zbl 0641.10026)] J.-P. Serre proposed a refined version of this conjecture in which he specified a level \(N(\rho)\) and a weight \(k(\rho)\) for the conjectured modular form. This has become known as the “Refined Serre conjecture”.
The paper under review is a report on the status of the following question: to which extent does the Serre conjecture imply the refined Serre conjecture? In other words, if we assume that a representation does arise from a modular form, does it follow that it arises from a form of the predicted level and weight? This is a fundamental question whose answer plays a fundamental role, for example, in the recent work of K. Ribet and of A. Wiles on Fermat’s last theorem and the Shimura-Taniyama conjecture.
The paper reports on work of many mathematicians, including N. Boston, H. Carayol, F. Diamond, B. Edixhoven, G. Faltings, B. H. Gross, B. Jordan, H. W. Lenstra, R. Livné, B. Mazur, K. Ribet, and J.-P. Serre, leading to an almost complete (positive) answer to this question. (The missing piece has since been supplied by F. Diamond in “The refined conjecture of Serre”, in Elliptic curves, modular forms, and Fermat’s last theorem, ed. by J. Coates and S. T. Yau, International Press. Ser. Number Theory 1, 22–37 (1995; Zbl 0853.11031).)
In addition, the last paper includes a new result on “lowering the level” from \(Mp\) to \(M\) (where \(p\) is a prime number not dividing \(M\ell\)) that does not depend on assumptions about the divisibility of \(M\) by \(\ell\). Finally, the author points out exactly what was then missing (and has since been supplied by Diamond) for a full solution of the problem (for odd primes \(\ell\)).
This is a significant paper, both for its original contributions and for the way it organizes and interprets the project of showing that the weak Serre conjecture implies the refined Serre conjecture.
For the entire collection see [Zbl 0788.00054].

MSC:

11F11 Holomorphic modular forms of integral weight
11F80 Galois representations
11R32 Galois theory
11G18 Arithmetic aspects of modular and Shimura varieties
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