2 results
Eisenstein cocycles in motivic cohomology
- Part of
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- 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
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- Article
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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.
On n-Dimensional Steinberg Symbols
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- Journal:
- Canadian Journal of Mathematics / Volume 60 / Issue 6 / 01 December 2008
- Published online by Cambridge University Press:
- 20 November 2018, pp. 1387-1405
- Print publication:
- 01 December 2008
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The aim of this work is to provide a new approach for constructing
$n$-dimensional Steinberg symbols on discrete valuation fields from 
$\left( n\,+\,1 \right)$-cocycles and to study reciprocity laws on curves related to these symbols.
