Constructing elliptic curves from Galois representations

Given a non-isotrivial elliptic curve over an arithmetic surface, one obtains a lisse $\mathrm{\ell }$-adic sheaf of rank two over the surface. This lisse sheaf has a number of straightforward properties: cyclotomic determinant, finite ramification, rational traces of Frobenius, and somewhere not potentially good reduction. We prove that any lisse sheaf of rank two possessing these properties comes from an elliptic curve.