Five-dimensional compatible systems and the Tate conjecture for elliptic surfaces

Let (ρ_λ GQ→ GL₅(Ē_λ))_λ be a strictly compatible system of Galois representations such that no Hodge--Tate weight has multiplicity $5$. We show that if ρλ₀ is irreducible for some λ₀, then ρ_λ is irreducible for all but finitely many λ. More generally, if (ρ_λ)_λ is essentially self-dual, we show that either ρ_λ is irreducible for all but finitely many λ, or the compatible system (ρ_λ)_λ decomposes as a direct sum of lower-dimensional compatible systems. We apply our results to study the Tate conjecture for elliptic surfaces. For example, if X₀ y² + (t+3)xy + y= x³, we prove the codimension one -adic Tate conjecture for all but finitely many , for all but finitely many general, degree $3$, genus $2$ branched multiplicative covers of X₀. To prove this result, we classify the elliptic surfaces into six families, and prove, using perverse sheaf theory and a result of Cadoret--Tamagawa, that if one surface in a family satisfies the Tate conjecture, then all but finitely many do. We then verify the Tate conjecture for one representative of each family by making our irreducibility result explicit: for the compatible system arising from the transcendental part of H²ₑₜ(XQ̄, Q₍1)) for a representative X, we formulate an algorithm that takes as input the characteristic polynomials of Frobenius, and terminates if and only if the compatible system is irreducible.