entire critical points of the abelian higgs functional are known to blow down to generalized minimal submanifolds (of codimension 2). in this talk we prove an allard type large-scale regularity result for the zero set of solutions. in the "multiplicity one" regime, we show the uniqueness of blow-downs and classify entire solutions in low dimensions and minimizers in all dimensions; thus obtaining an analogue of savin's theorem in codimension two. this is based on a joint work with guido de philippis and alessandro pigati.