Beilinson has conjectured that for a regular, complete, geometrically irreducible curve $C$ of genus $g$ over a number field $k$, $rank(K_2(C))=g[k:\mathbb{Q}]$. As far as I know it is not known in the general case whether $rank(K_2(C))\geq g[k:\mathbb{Q}]$ or not. I'm looking for examples of smooth projective planar curves of genus $g\geq 3$ over $k$, which is known there are at least $g[k:\mathbb{Q}]$ independent elements in the second algebraic $K$group. (In particular the curve isn't hyperelliptic)
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4$\begingroup$ Quite a lot is known about Fermat curves; the paper of Otsubo "On the regulator of Fermat motives and generalized hypergeometric functions" might be a useful place to start. $\endgroup$– nafSep 3, 2018 at 4:58

2$\begingroup$ Modular curves $X_1(N) / \mathbf{Q}$ are another natural thing to try: the $K_2$ of these curves is rather well understood thanks to Bloch, Beilinson and Kato. $\endgroup$– David LoefflerSep 3, 2018 at 6:22

5$\begingroup$ Rob de Jeu and Hang Liu have a paper 'On K_2 of certain families of curves', International Mathematics Research Notices, 2015 (2015), issue 21, 1092910958, arxiv.org/abs/1402.4822. They give examples of the type that you are looking for, I believe. $\endgroup$– Tim DokchitserSep 3, 2018 at 8:10
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