ℓ-class groups of fields in Kummer towers

• Autores: Jianing Li, Yi Ouyang, Yue Xu, Shenxing Zhang
• Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 66, Nº 1, 2022, págs. 235-267
• Idioma: catalán
• Títulos paralelos:
  • ℓ-class groups of fields in Kummer towers
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• Resumen

  • Let ℓ and b be prime numbers and Kn,m = Q(p1 n , ζ2`m). We study the `-class group of Kn,m in this paper. When ` = 2, we determine the structure of the 2-class group of Kn,m for all (n, m) ∈ Z 2 ≥0 in the case p ≡ 3, 5 mod 8, and for (n, m) = (n, 0), (n, 1), or (1, m) in the case p ≡ 7 mod 16, generalizing the results of Parry about the 2-divisibility of the class number of K2,0. We also obtain results about the `-class group of Kn,m when ` is odd and in particular when ` = 3. The main tools we use are class field theory, including Chevalley’s ambiguous class number formula and its generalization by Gras, and a stationary result about the `-class groups in the 2-dimensional Kummer tower {Kn,m}.