Galois representations congruent to those coming from Shimura varieties. (English) Zbl 0823.11022

Tilouine, J.

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, 625-638 (1994).

The author surveys various methods for constructing a compatible system of \(\ell\)-adic representations \(\sigma (\pi)= (\sigma (\pi)_ \ell )_ \ell\) associated to an arithmetic automorphic representation \(\pi\). In the general case, the existence of such a system of representations would follow from the existence of a motive associated to the automorphic representation. The methods described here do not construct such a motive, but rather obtain the representations directly, essentially by using congruences between automorphic representations. (The correct notion of congruence must be defined in each case, the crucial thing being that congruence on the automorphic side must imply congruences on the Galois representations side.)
The basic scheme is as follows. One first proves the existence of the system \(\sigma (\pi')\) for a large class of arithmetic automorphic representations \(\pi'\). Then, for any given arithmetic automorphic representation \(\pi\), one produces infinitely many representations \(\pi'\) which are congruent to \(\pi\). Finally, one obtains \(\sigma (\pi)\) by “patching together” all the \(\sigma (\pi')\). This scheme gives a very illuminating way of understanding a number of important results.
The article discusses three ways to realize this scheme, which depend on different ways to define “congruence”. First, one can use congruence modulo infinitely many primes \(p\in \mathbb{Z}\). This is the case in the construction of the Galois representations attached to eigenforms of weight 1 by P. Deligne and J.-P. Serre [Ann. Sci. École Norm. Supér., IV. Sér. 7 (1974), 507–530 (1975; Zbl 0321.10026))] and in the analogous construction by J. D. Rogawski and J. B. Tunnell [Invent. Math. 74, 1–42 (1983; Zbl 0523.12009)] for Hilbert modular forms.
Second, one can consider congruence modulo infinitely many prime ideals of the Iwasawa algebra \(\Lambda= \mathbb{Z}_p [[T]]\). This is based on ideas of Hida, and was used by A. Wiles [Invent. Math. 94, 529–573 (1988; Zbl 0664.10013)] to give a different construction in the Hilbert modular case, which removes, in the \(p\)-ordinary case, a technical assumption in the Rogawski-Tunnel theorem. (The author describes each of these methods as using “horizontal” congruences, either in \(\mathrm{Spec}(\mathbb{Z})\) or in \(\mathrm{Spec}(\Lambda)\).)
Finally, one can consider “vertical” congruences, where one works modulo \(p^n\) for varying \(n\). This is the method used by R. Taylor to construct Galois representations both in the Hilbert modular case [Invent. Math. 98, 265-280 (1989; Zbl 0705.11031)] and in the case of forms over \(\mathrm{GSp}(4, \mathbb{Q})\) [Duke Math. J. 63, 281–332 (1991; Zbl 0810.11033)].
For the entire collection see [Zbl 0788.00054].

Reviewer: F. Gouvêa (Waterville)

MSC:

11F70	Representation-theoretic methods; automorphic representations over local and global fields
14G35	Modular and Shimura varieties
11F11	Holomorphic modular forms of integral weight

Keywords:

\(\ell\)-adic representations; arithmetic automorphic representation; Galois representations

Citations:

Zbl 0321.10026; Zbl 0523.12009; Zbl 0664.10013; Zbl 0705.11031; Zbl 0810.11033