We construct families of smooth, proper, algebraic curves in characteristic 0, of arbitrary genus g, together with g elements in the kernel of the tame symbol. We show that those elements are in general independent by a limit calculation of the regulator. Working over a number field, we show that in some of those families the elements are integral. We determine when those curves are hyperelliptic, finding, in particular, that over any number field we have non-hyperelliptic curves of all composite genera g with g independent integral elements in the kernel of the tame symbol. We also give families of elliptic curves over real quadratic fields with two independent integral elements.where · · · denotes the subgroup generated by the indicated elements. The class of a ⊗ b is denoted {a, b}, so that K 2 (F ) is an Abelian group (written additively), with generators {a, b} for a and b in F * , and relationsThese relations also imply {a, −a} = 0 and {a, b} = −{b, a}.
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.
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