Analogues of Iwasawa’s μ=0 conjecture and the weak Leopoldt conjecture for a non-cyclotomic Z_2 -extension

Junhwa Choi School of Mathematics, Korea Institute for Advanced Study, 85 Hoegi-ro, Dongdaemun-gu, Seoul 02455, Republic of Korea Yukako Kezuka Fakultät für Mathematik, Universität Regensburg, Germany Yongxiong Li Yau Mathematical Sciences Center, Tsinghua University, Beijing, China

Number Theory mathscidoc:2204.24006

Asian Journal of Mathematics, 23, 383-400, 2019.7
Let K=Q(√(-q)), where q is any prime number congruent to 7 modulo 8, and let O be the ring of integers of K. The prime 2 splits in K, say 2O=pp∗, and there is a unique Z_2-extension K_∞ of K which is unramified outside p. Let H be the Hilbert class field of K, and write H_∞=HK_∞. Let M(H_∞) be the maximal abelian 2-extension of H_∞ which is unramified outside the primes above p, and put X(H_∞)=Gal(M(H_∞)/H_∞). We prove that X(H_∞) is always a finitely generated Z_2-module, by an elliptic analogue of Sinnott’s cyclotomic argument. We then use this result to prove for the first time the weak p-adic Leopoldt conjecture for the compositum J_∞ of K_∞ with arbitrary quadratic extensions J of H. We also prove some new cases of the finite generation of the Mordell–Weil group E(J_∞) modulo torsion of certain elliptic curves E with complex multiplication by O.
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@inproceedings{junhwa2019analogues,
  title={Analogues of Iwasawa’s μ=0 conjecture and the weak Leopoldt conjecture for a non-cyclotomic Z_2 -extension},
  author={Junhwa Choi, Yukako Kezuka, and Yongxiong Li},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220422145450564533106},
  booktitle={Asian Journal of Mathematics},
  volume={23},
  pages={383-400},
  year={2019},
}
Junhwa Choi, Yukako Kezuka, and Yongxiong Li. Analogues of Iwasawa’s μ=0 conjecture and the weak Leopoldt conjecture for a non-cyclotomic Z_2 -extension. 2019. Vol. 23. In Asian Journal of Mathematics. pp.383-400. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220422145450564533106.
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