On the $K_4$ group of modular curves

Abstract

We construct elements in the $K_4$ group of modular curves using the polylogarithmic complexes of weight 3 defined by Goncharov and De Jeu. The construction is uniform in the level and relies on new modular units arising as cross-ratios of division values of the Weierstrass $\wp$ function. These units provide explicit triangulations of the $3$-term relations in $K_2$, which in turn give rise to elements in $K_4$. Based on numerical computations and on results of Wang, we conjecture that these elements are proportional to the Beilinson elements defined via Eisenstein symbols.

40 pages. Significant revision. I rewrote the introduction and restructured the paper, with some mathematical improvements. I also included a running example with the modular curve X_1(15)

References in corpus (2)

• Regulators of Siegel units and applications
• Motivic correlators, cluster varieties and Zagier's conjecture on zeta(F,4)

Cited by in corpus (1)

• Eisenstein cocycles in motivic cohomology