Weber’s Class Number One Problem
Let p and ℓ be prime numbers. In this paper, we consider the ℓ-indivisibility of the class numbers of the intermediate fields in the cyclotomic Z p -extension of Q. Moreover, we study the class numbers of the intermediate fields in the composite of such extensions.
The authors thank the organizers of Iwasawa 2012 for giving us the opportunity to talk. The authors also thank the referee for reading this paper carefully and offering several invaluable suggestions.
- Aoki, M., Fukuda, T.: An algorithm for computing p-class groups of abelian number fields.In: Hess, F., Pauli, S., Pohst, M. (eds.) Algorithmic Number Theory. Lecture Notes in Computer Science, vol. 4076, pp. 56–71. Springer, Berlin/Heidelberg (2006)Google Scholar
- Coates, J.: The enigmatic Tate-Shafarevich group. In: Fifth International Congress of Chinese Mathematicians. Part 1, 2, AMS/IP Studies in Advanced Mathematics, vol. 51, Part. 1, 2, pp. 43–50. American Mathematical Society, Providence (2012)Google Scholar
- Fukuda,T., Komatsu, K.: Weber’s class number problem in the cyclotomic Z 2-extension of Q. Exp. Math. 18(2), 213–222 (2009)Google Scholar
- Horie, K: The ideal class group of the basic Z p-extension over an imaginary quadratic field. Tohoku Math. J. (2) 57, 375–394 (2005b)Google Scholar
- Morisawa, T., Okazaki, R.: Mahler measure and Weber’s class number problem in the cyclotomic Z p-extension of Q for odd prime number p. Tohoku Math. J. 65(2), 253–272 (2013)Google Scholar
- Okazaki, R.: On a lower bound for relative units, Schinzel’s lower bound and Weber’s class number problem (preprint)Google Scholar
- Washington, L.C.: Introduction to Cyclotomic Fields. Graduate Texts in Mathematics, vol. 83, 2nd edn. Springer, New York/Heidelberg/Berlin (1997)Google Scholar