${\bm K}_{\bm 2}$ of four families of (hyper-) elliptic curves

Abstract

We construct elements in the K2 group on four families of (hyper-) elliptic curves of arbitrary genus g. If the curves are defined over number fields, we show that these elements are integral under certain condition. We prove that some of these elements are linearly independent for suitable parameters. For the special case of g = 1, we show that all the elements are linearly independent modulo some universal relations for certain parameters. As an application, we give families of elliptic curves with two explicit independent integral elements over two real quadratic fields respectively.