$L_{\kappa\omega}$-equivalence of abelian groups with partial decomposition bases

We consider the class of abelian groups possessing partial decomposition bases in the language L_{\kappa \omega} for uncountable cardinals \kappa . Jacoby, Leistner, Loth and Strüngmann developed a numerical invariant deduced from the classical global Warfield invariant and proved that if two groups have identical modified Ulm invariants and Warfield invariants up to \omega\delta for some ordinal \delta , then they are equivalent in L_{\infty\omega}^{\delta} . Subsequently, Jacoby and Loth showed that the converse is true for appropriate \delta . In this paper we prove that the modified Warfield invariant up to \kappa is expressible in L_{\kappa\omega} , thus a complete classification theorem in L_{\kappa \omega} is obtained. This generalizes a result of Barwise and Eklof.