On the K_4 group of modular curves

Sur le groupe K_4 des courbes modulaires

François Brunault (1, 2)

1 ENS Lyon - École normale supérieure - Lyon
2 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

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

Domains: Mathematics [math]; Mathematics [math]; Number Theory [math.NT]; Mathematics [math]; K-Theory and Homology [math.KT]

François Brunault. On the K_4 group of modular curves. 2022. ⟨ensl-03012466v2⟩