Compatible system of Galois representations

Let $F$ and $E$ be number fields. A weakly compatible system of $n$ -dimensional Galois representations over $E$ consists of continuous semisimple representations

\rho_\lambda: \operatorname{Gal}(\overline F/F) \longrightarrow \operatorname{GL}_n(\overline E_\lambda)

of the absolute Galois group, for finite places $\lambda$ of $E$ , together with a finite set $S$ of places of $F$ , such that for every $v\notin S$ and every $\lambda\nmid v$ , the restriction of $\rho_\lambda$ to the inertia subgroup at $v$ is trivial, and

P_v(X)= \det\!\left(1-X\,\rho_\lambda(\operatorname{Frob}_v)\right) \in E[X]

is independent of $\lambda$ , after the fixed embeddings of $E$ into the local coefficient fields.

Frobenius convention

The polynomial changes by inverting eigenvalues when arithmetic Frobenius is replaced by geometric Frobenius. A compatible system must use one convention consistently. Many arithmetic sources use geometric Frobenius in local class field theory but arithmetic Frobenius in étale-cohomological characteristic polynomials; the formula, not the word alone, is decisive.

Stronger compatibility

The adjective “compatible system” is used with several strengths. A stronger system can additionally prescribe:

a $\lambda$ -independent multiset of Hodge–Tate numbers at each embedding;
de Rham or crystalline behavior at places above $\ell$ ;
compatible Weil–Deligne representations at all finite places;
purity of a fixed weight;
a common coefficient field and polarization.

These conditions should be listed rather than inferred from the name.

Semisimplicity and ramification

Almost-all Frobenius polynomials determine only the semisimplification, by the Chebotarev density theorem. They do not by themselves recover monodromy operators at ramified places. Full local–global compatibility contains strictly more information.

Automorphic origin

Regular algebraic cuspidal automorphic representations of general linear groups produce compatible systems in many theorem-level settings. The general expectation for reductive groups is naturally $\widehat G$ - or $L$ -group-valued and requires a representation of the dual group to recover the displayed $\operatorname{GL}_n$ -valued system.

References

Jean-Pierre Serre, Abelian $\ell$ -Adic Representations and Elliptic Curves, second edition, 1998.
Thomas Barnet-Lamb, Toby Gee, David Geraghty, and Richard Taylor, “Potential automorphy and change of weight,” §5. arXiv.