On the K_4 group of modular curves

Sur le groupe K_4 des courbes modulaires

François Brunault (1)

1 ENS de Lyon - École normale supérieure de Lyon

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

Domaines: Mathématiques [math]; Théorie des nombres [math.NT]; K-théorie et homologie [math.KT]

François Brunault. On the K_4 group of modular curves. 2020. ⟨ensl-03012466v1⟩