Geometric Galois representations. (English) Zbl 0839.14011

Fontaine, Jean-Marc; Mazur, Barry

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, 41-78 (1995).

The paper deals with \(p\)-adic representations of the absolute Galois-group of a number field (and also a \(p\)-adic local field). These are called “geometric” if they are unramified at almost all primes and potentially semistable at the others. For primes dividing \(p\) “semistable” is defined via crystalline theory. It is conjectured that irreducible geometric representations “come from geometry”, that is somehow occur in the cohomology of algebraic varieties. There are also conjectures about finiteness of isomorphism classes (under bounds on the ramification) and deformations. The second half of the paper studies in great detail 2-dimensional geometric \(p\)-adic representations, and classifies their restriction to the decomposition-group at a place dividing \(p\).
For the entire collection see [Zbl 0824.00025].

Reviewer: G.Faltings (Bonn)

MSC:

14F30	\(p\)-adic cohomology, crystalline cohomology
11R32	Galois theory
11S20	Galois theory

Keywords:

\(p\)-adic representations of the absolute Galois-group; 2-dimensional geometric \(p\)-adic representations; crystalline theory