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arXiv · 2501.11277

Some Kummer extensions over maximal cyclotomic fields, a finiteness theorem of Ribet and TKND-AVKF fields

Takahiro Murotani·Yoshiyasu Ozeki
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Abstract

It is a theorem of Ribet that an abelian variety defined over a number field $K$ has only finitely many torsion points with values in the maximal cyclotomic extension field $K^{\mathrm{cyc}}$ of $K$. Recently, R\"ossler and Szamuely generalized Ribet's theorem in terms of the \'etale cohomology with $\mathbb{Q}/\mathbb{Z}$-coefficients of a smooth proper variety. In this paper, we show that the same finiteness holds even after replacing $K^{\mathrm{cyc}}$ with the field obtained by adjoining to $K$ all roots of all elements of a certain subset of $K$. Furthermore, we give some new examples of TKND-AVKF fields; the notion of TKND-AVKF is introduced by Hoshi, Mochizuki and Tsujimura, and TKND-AVKF fields are expected as one of suitable base fields for anabelian geometry.

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Takahiro Murotani, Yoshiyasu Ozeki. 2025-01-20. Some Kummer extensions over maximal cyclotomic fields, a finiteness theorem of Ribet and TKND-AVKF fields. https://arxiv.org/abs/2501.11277

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