On K_2 of certain families of curves

TL;DR

This paper constructs and analyzes families of algebraic curves over number fields, demonstrating the existence of independent elements in the kernel of the tame symbol and exploring their properties, including integrality and hyperellipticity, across various genera.

Contribution

It introduces a method to produce families of curves with independent kernel elements in the tame symbol, extending understanding of their algebraic and geometric properties.

Findings

Existence of independent kernel elements in constructed families

Identification of conditions for integrality over number fields

Characterization of hyperelliptic versus non-hyperelliptic curves

Abstract

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.