Lifting Galois representations and a conjecture of Fontaine and Mazur

Abstract

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.