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

Let $(ρ_λ\\colon G_{\\mathbb Q}\\to \\operatorname{GL}_5(\\overline{E}_λ))_λ$ be a strictly compatible system of Galois representations such that no Hodge--Tate weight has multiplicity $5$. Under mild assumptions, we show that if $ρ_{λ_0}$ is irreducible for some $λ_0$, then $ρ_λ$ is irreducible for all but finitely many priimes $λ$. 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 $X_0\\colon y^2 + (t+3)xy + y= x^3$, we prove the codimension one $\\ell$-adic Tate conjecture for all but finitely many $\\ell$, for all but finitely many general, degree $3$, genus $2$ branched multiplicative covers of $X_0$. To prove this result, we classify the elliptic surfaces into four one-dimensional families and two isolated classes. For each of the four families, we prove, using perverse sheaf theory and a result of Cadoret--Tamagawa, that if the relevant $5$-dimensional Galois representation is irreducible for one surface in a family, then it is irreducible for all but finitely many surfaces in that family. We then verify this irreducibility for one representative of each family by making our irreducibility result explicit: for the compatible system arising from the transcendental part of $H^2_{\\mathrm{et}}(X_{\\overline{\\mathbb Q}}, \\mathbb{Q}_\\ell(1))$ 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.