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

Let $({\rho }_{\lambda }\text{\hspace{0.17em}⁣}:G_{\mathbb{Q}}\to {\mathrm{GL}}_5({\bar{E}}_{\lambda }){)}_{\lambda }$ be a strictly compatible system of Galois representations such that no Hodge–Tate weight has multiplicity $5$. Under mild assumptions, we show that if ${\rho }_{{\lambda }_0}$ is irreducible for some ${\lambda }_0$, then ${\rho }_{\lambda }$ is irreducible for all but finitely many priimes $\lambda$. More generally, if $({\rho }_{\lambda }{)}_{\lambda }$ is essentially self-dual, we show that either ${\rho }_{\lambda }$ is irreducible for all but finitely many $\lambda$, or the compatible system $({\rho }_{\lambda }{)}_{\lambda }$ 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 $X_0\text{\hspace{0.17em}⁣}:y^2+(t+3)xy+y=x^3$, we prove the codimension one $\mathrm{\ell }$-adic Tate conjecture for all but finitely many $\mathrm{\ell }$, 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 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 $H_{\mathrm{e}\mathrm{t}}^2(X_{\bar{\mathbb{Q}}},{\mathbb{Q}}_{\mathrm{\ell }}(1))$ 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.