$n$-Valued Analogue of Abel's Theorem

Abstract

Abel's classical theorem shows that a continuously differentiable one-dimensional associative law with a two-sided unit is locally equivalent to addition under a change of coordinate. We prove an $n$-valued analogue in the algebraic and formal settings. Over a field of characteristic zero, every associative algebraic $n$-valued monoid on ${\mathbb{P}}^1$ with a strong two-sided unit is commutative. Over a reduced commutative $\mathbb{Q}$-algebra, any one-dimensional formal $n$-valued group is commutative. We then classify $n$-valued formal groups over arbitrary $\mathbb{Q}$-algebras that are integral domains: for any $n$-valued formal group there is a unique divisor $m$ of $n$ such that the law is strongly isomorphic to the $n/m$-fold diagonal of the cyclic elementary $m$-valued law ${\mathbb{G}}_m$. This proves Kholodov's classification conjecture over integral domain $\mathbb{Q}$-algebras.