On formulas for the index of the circular distributions

Abstract

A circular distribution is a Galois equivariant map ψ from the roots of unity μ _∞ to an algebraic closure of ${\mathbb{Q}}$ such that ψ satisfies product conditions, ${\prod_{\zeta^{d} = \epsilon}\psi(\zeta) = \psi(\epsilon)}$ for ϵ ∈ μ _∞ and ${d \in \mathbb{N}}$ , and congruence conditions for each prime number l and ${s \in \mathbb{N}}$ with (l, s) = 1, ${ \psi(\epsilon \zeta) \equiv \psi(\zeta)}$ modulo primes over l for all ${\epsilon\in\mu_{l}, \zeta \in \mu_{s}}$ , where μ _l and μ _s denote respectively the sets of lth and sth roots of unity. For such ψ, let ${P^\psi_s}$ be the group generated over ${\mathbb{Z}[\mbox{Gal} ({\mathbb{Q}}(\mu_{s})/{\mathbb{Q}})]}$ by ${\psi(\zeta), \zeta \in \mu_{s}}$ and let ${C^\psi_s}$ be ${P^\psi_s \bigcap U_s}$ , where U _s denotes the global units of ${\mathbb{Q}(\mu_s)}$ . We give formulas for the indices ${(P_s:P^\psi_s)}$ and ${(C_s : C^\psi_s)}$ of ${P^\psi_s}$ and ${C^\psi_s}$ inside the circular numbers P _s and units C _s of Sinnott over ${\mathbb{Q}(\mu_s)}$ .