Polymultiplicative maps associated with the algebra of Iterated Laurent series and the higher-dimensional Contou-Carrere Symbol

TL;DR

This work addresses the rigidity of higher-dimensional Contou-Carrère symbols by classifying polymultiplicative, functorial maps from iterated Laurent series units to the base ring that are invariant under continuous automorphisms. The authors develop an additive analogue via a residue map, proving that any such invariant multilinear form on $\mathcal{L}^n(\mathbb{G}_a)$ is a scalar multiple of the residue of a differential form. Building on this, they show that any invariant polymultiplicative symbol on $\mathcal{L}^n(\mathbb{G}_m)$ factors as an $m$-th power of $CC_n$, twisted by a valuation-dependent factor encoded by a polylinear map $\omega$ on $\underline{\mathbb{Z}}^n_R$, i.e. $(f_0,\dots,f_n) = \omega(v(f_0),\dots,v(f_n)) CC_n(f_0,\dots,f_n)^m$. The results unify and extend the known one- and two-dimensional cases, provide a structural description of continuous automorphisms, and establish a rigorous framework for the uniqueness of higher-dimensional reciprocity symbols under symmetry constraints.

Abstract

We study functorial polymultiplicative maps from the multiplicative group of the algebra of $n$-times iterated Laurent series over a commutative ring in $n+1$ variables into the multiplicative group of the ring. It is proven that if such a map is invariant under continuous automorphisms of this algebra, then it coincides, up to a sign, with an integer power of the $n$-dimensional Contou-Carrère symbol.

Theorems & Definitions (57)

• Theorem 1
• Proposition 1
• Definition 1
• Proposition 2: GorOsi15, Proposition 4.9
• Proposition 3: GorOsi15, Proposition 8.15
• Remark 2.1
• Theorem 2: GorOsi15, Theorem 8.17, Proposition 8.22
• Theorem 3: GorOsiCont, Theorem 6.8
• Theorem 4
• Lemma 3.1