1 results
Eisenstein cocycles in motivic cohomology
- Part of
-
- Journal:
- Compositio Mathematica / Volume 160 / Issue 10 / October 2024
- Published online by Cambridge University Press:
- 30 October 2024, pp. 2407-2479
- Print publication:
- October 2024
-
- Article
-
- You have access
- HTML
- Export citation
-
Several authors have studied homomorphisms from first homology groups of modular curves to
$K_2(X)$, with 
$X$ either a cyclotomic ring or a modular curve. These maps send Manin symbols in the homology groups to Steinberg symbols of cyclotomic or Siegel units. We give a new construction of these maps and a direct proof of their Hecke equivariance, analogous to the construction of Siegel units using the universal elliptic curve. Our main tool is a 
$1$-cocycle from 
$\mathrm {GL}_2(\mathbb {Z})$ to the second 
$K$-group of the function field of a suitable group scheme over 
$X$, from which the maps of interest arise by specialization.
