Summary: Mumford has constructed 4-dimensional abelian varieties with trivial endomorphism ring, but whose Mumford--Tate group is much smaller than the full symplectic group. We consider such an abelian variety, defined over a number field $F$, and study the associated $p$-adic Galois representation. For $F$ sufficiently large, this representation can be lifted to $\mathbf{G}_m(\mathbf{Q}_p)\times\mathrm{SL}_2(\mathbf{Q}_p)^3$. Such liftings can be used to construct Galois representations which are geometric in the sense of a conjecture of Fontaine and Mazur. The conjecture in question predicts that these representations should come from algebraic geometry. We confirm the conjecture for the representations constructed here.

11G10, 11F80, 14K15

geometric Galois representation, fontaine--Mazur conjecture, motives