Abelian surfaces are complex tori whose enumerative invariants satisfy remarkable regularity properties. The computation of their (reduced) Gromov-Witten invariants for the so called primitive classes is fairly well understood and many complete computations are available. A few years ago, G. Oberdieck conjectured a multiple cover formula expressing in a very simple way the invariants for the non-primitive classes in terms of the primitive one. The proof of the conjecture would solve completely the GW theory of abelian surfaces. In this talk, I will explain a proof of multiple cover formula conjecture for many insertions. The argument relies on a reduced degeneration formula and on the computation of correlated Gromov-Witten invariants for trivial bundles on elliptic curves. This is joint work with T. Blomme.