On formulas for the index of the circular distributions

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Abstract

A circular distribution is a Galois equivariant map ψ from the roots of unity μ to an algebraic closure of ℚ such that ψ satisfies product conditions, ∏ζd =ε ψ (ζ) = ψ(ε) for ε ∈ μ and d ∈ ℕ, and congruence conditions for each prime number l and s ∈ ℕ with (l, s) = 1, ψ(ε ζ) ≡ ψ(ζ) modulo primes over l for all ε ∈ μl, ζ ∈ μs, where μl and μ s denote respectively the sets of lth and sth roots of unity. For such ψ, let Pψs be the group generated over ℤ[Gal(ℚ(μs)/ℚ)] by ψ(ζ), ζ ∈ μs and let Cψs be Pψs ∩ Us, where Us denotes the global units of ℚ(μs). We give formulas for the indices (Ps:P ψs) and (Cs : Cψ s) of Pψs and Cψs inside the circular numbers Ps and units Cs of Sinnott over ℚ(μs).

Original languageEnglish
Pages (from-to)381-396
Number of pages16
JournalManuscripta Mathematica
Volume127
Issue number3
DOIs
Publication statusPublished - 2008 Nov 1

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All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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