Elements in $K_4$ and regulator maps of Fermat curves

We construct explicit elements in the group $K_4^{(3)}$ of the Fermat curves $x^N+y^N=1$ for all $N\ge 3$. The construction, which is uniform in $N$, uses polylogarithmic complexes and a map of de Jeu to $K$-theory. We prove that the elements are non-trivial by showing that their images under Beilinson’s regulator map are non-zero. Notably, we obtain explicit formulas for their regulator integrals involving special values of Zagier’s trilogarithm function. As a corollary, we show that these regulator integrals are asymptotic to $\frac{3}{2}\zeta (3)N^2$ as $N\to +\mathrm{\infty }$. Moreover, we derive formulas for the regulators of our elements in terms of hypergeometric functions, generalizing results of Otsubo for $K_2$ groups of Fermat curves. Finally, we numerically verify some cases of Beilinson’s conjectures on special values of $L$-functions at $s=3$ for $N\in \{3,4,6\}$.