  This paper is the first part in a series of three papers devoted to the study
of enumerative invariants of abelian surfaces through the tropical approach. In
this paper, we consider the enumeration of genus $g$ curves of fixed degree
passing through $g$ points. We compute the multiplicity provided by a
correspondence theorem due to T. Nishinou and show that it is possible to
refine this multiplicity in the style of Block-G\"ottsche to get tropical
refined invariants.
