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 language | English |
|---|---|
| Pages (from-to) | 381-396 |
| Number of pages | 16 |
| Journal | Manuscripta Mathematica |
| Volume | 127 |
| Issue number | 3 |
| DOIs | |
| Publication status | Published - 2008 Nov 1 |
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All Science Journal Classification (ASJC) codes
- Mathematics(all)
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On formulas for the index of the circular distributions. / Seo, Soogil.
In: Manuscripta Mathematica, Vol. 127, No. 3, 01.11.2008, p. 381-396.Research output: Contribution to journal › Article
TY - JOUR
T1 - On formulas for the index of the circular distributions
AU - Seo, Soogil
PY - 2008/11/1
Y1 - 2008/11/1
N2 - 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).
AB - 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).
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U2 - 10.1007/s00229-008-0214-7
DO - 10.1007/s00229-008-0214-7
M3 - Article
VL - 127
SP - 381
EP - 396
JO - Manuscripta Mathematica
JF - Manuscripta Mathematica
SN - 0025-2611
IS - 3
ER -