Functional equations of polylogarithms in motivic cohomology
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Abstract
For an infinite
field F, we study the integral relationship between the Bloch group B_2(F) and the higher Chow group CH^2(F,3) by proving some relations corresponding to the functional equations of the dilogarithm. As a second result, the groups involved in Suslin’s exact sequence
0 → Tor^1(F^× ,F^×)∼ → CH^2(F,3) → B_2(F) → 0
are identified with homology groups of the cycle complex Z^2(F,•) computing Bloch’s higher Chow groups.
Using these results, we give explicit cycles in motivic cohomology generating the integral motivic cohomology groups of some specific number fields and determine whether a given
cycle in the Chow group already lives in one of the other groups of Suslin’s sequence. In principle, this enables us to find a presentation of the codimension two Chow group of an
arbitrary number field.
Finally, we also prove some relations in the higher Chow groups of codimension three modulo 2-torsion coming from relations in the higher Bloch group B_3(F) modulo 2-torsion. Further, we can prove
a series of relations in CH^ 3(Q(zeta_p),5) for a primitive pth root of unity zeta_p.