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

Abstract

Let $(\ddot{I}{}_{\hat{I}}»:G_{\mathbb{Q}}\to {\mathrm{GL}}_5({\overline{E}}_{\hat{I}}»){)}_{\hat{I}}»$ be a strictly compatible system of Galois representations such that no Hodge--Tate weight has multiplicity $5$. Under mild assumptions, we show that if $\ddot{I}{}_{\hat{I}{»}_0}$ is irreducible for some $\hat{I}{»}_0$, then $\ddot{I}{}_{\hat{I}}»$ is irreducible for all but finitely many priimes $\hat{I}»$. More generally, if $(\ddot{I}{}_{\hat{I}}»{)}_{\hat{I}}»$ is essentially self-dual, we show that either $\ddot{I}{}_{\hat{I}}»$ is irreducible for all but finitely many $\hat{I}»$, or the compatible system $(\ddot{I}{}_{\hat{I}}»{)}_{\hat{I}}»$ 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 $X_0:y^2+(t+3)xy+y=x^3$, 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 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 $H_{\mathrm{et}}^2(X_{\overline{\mathbb{Q}}},{\mathbb{Q}}_{\ell }(1))$ 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.

57 pages. Numerous corrections following peer review. Comments welcome!