On ? of algebraic monodromy groups in compatible systems of representations

Abstract

Consider a profinite group G, and a collection of continuous representations ρ` : G → GLn(Q`), indexed by a set L of rational primes `. Suppose that G is endowed with a dense subset of “Frobenius” elements {Fα|α ∈ A}. The system {ρ`} is called a compatible system of `-adic representations if, for every α ∈ A, the characteristic polynomial of ρ`(Fα) has coefficients in the field of rational numbers and does not depend on `. In 6.5, we will give a more precise and less restrictive definition which allows us to throw out some bad pairs (`, α) in order to accomodate ramification. Our notion slightly generalizes Serre’s original definition [14]; to recover Serre’s definition, we take G to be the Galois group of a number field K and Fα to be Frobenius representatives for primes of K. Let G` be the Zariski closure of ρ`(G) in GLn,Q` . This is the algebraic monodromy group at `. Our question is the following: How does G` vary with `? One hopes for some kind of “`-independence.” At best, there can exist a global algebraic group G ⊂ GLn,Q such that every G` is conjugate to G ×Q Q`. Unfortunately, this does not always happen in the abstract setting in which we work, so we must settle for weaker `-independence results. We first recall what is already known in this direction. The compatibility condition bears only on the semisimple part of the elements ρ`(Fα), so we lose no information by assuming all ρ` to be semisimple representations. (Alternatively, we could take arbitrary representations and define G` as the quotient of the Zariski closure of ρ`(G) by its unipotent radical.) Thus {G`} is a family of reductive groups. Serre has proved