Triple symbols in arithmetic

Abstract

Triple symbols are arithmetic analogues of the mod $n$ triple linking number in topology, where $n>1$ is an integer. In this paper, we introduce a cohomological formulation of a mod $n$ triple symbol for characters over a number field containing a primitive $n$-th root of unity. Our definition is motivated by the arithmetic Chern--Simons theory and in this respect it differs from earlier approaches to triple symbols. We show that our symbol agrees with that of Rédei when $n=2$ and of Amano--Mizusawa--Morishita when $n=3$.

13 pages; in v2 (1) the reference [13] is added and (2) a minor error in Prop. 2 is corrected