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Kummer-faithful fields which are not sub-p-adic

Abstract

A Kummer-faithful field k is a perfect field such that the Kummer map associated to G is injective for every finite extension K of k and every semi-abelian variety G over K. A typical example of a Kummer-faithful field of characteristic zero is a sub-p-adic field for some prime number p. In particular, every number field is Kummer-faithful. In this paper, we give a construction of a Kummer-faithful field which is an infinite algebraic extension of the field of rational numbers. Moreover, we give some examples of Kummer-faithful fields which are not sub-p-adic by using the constructions.

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Acknowledgements

The author would like to express her sincere gratitude to Professor Yuichiro Taguchi who suggested her to consider this problem and gave her much valuable advice. She would like to express her appreciation to Professor Yoshiyasu Ozeki for his incisive comments on the proof of Theorem 2. She also thanks Professor Yuichiro Hoshi and Professor Kazuo Matsuno for meaningful discussions. She would like to offer her special thanks to Professor Toru Komatsu for permission to use his results in [8]. Finally, she is deeply grateful to the referees for their many helpful comments and constructive suggestions which improved the original manuscript.

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Correspondence to Sachiko Ohtani.

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Ohtani, S. Kummer-faithful fields which are not sub-p-adic. Res. number theory 8, 15 (2022). https://doi.org/10.1007/s40993-022-00311-2

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  • DOI: https://doi.org/10.1007/s40993-022-00311-2

Keywords

  • Kummer-faithful fields
  • Infinite algebraic extensions
  • PAC fields
  • Cyclotomic extensions
  • Cusp forms of level 1
  • Division points of an elliptic curve

Mathematics Subject Classification

  • 11R21
  • 14G05
  • 12E30