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The densities for 3-ranks of tame kernels of cyclic cubic number fields

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Abstract

Let F be a cubic cyclic field with t (⩾ 2) ramified primes. For a finite abelian group G, let r 3(G) be the 3-rank of G. If 3 does not ramify in F, then it is proved that t−1 ⩽ r 3(K 2 O F ) ⩽ 2t. Furthermore, if t is fixed, for any s satisfying t−1 ⩽ s ⩽ 2t−1, there is always a cubic cyclic field F with exactly t ramified primes such that r 3(K 2 O F ) = s. It is also proved that the densities for 3-ranks of tame kernels of cyclic cubic number fields satisfy a Cohen-Lenstra type formula

$$d_{\infty ,r} = 3^{ - r^2 } {{\prod\limits_{k = 1}^\infty {\left( {1 - 3^{ - k} } \right)} } \mathord{\left/ {\vphantom {{\prod\limits_{k = 1}^\infty {\left( {1 - 3^{ - k} } \right)} } {\prod\limits_{k = 1}^r {\left( {1 - 3^{ - k} } \right)} }}} \right. \kern-\nulldelimiterspace} {\prod\limits_{k = 1}^r {\left( {1 - 3^{ - k} } \right)} }}^2 .$$

This suggests that the Cohen-Lenstra conjecture for ideal class groups can be extended to the tame kernels of cyclic cubic number fields.

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Authors and Affiliations

  1. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    XiaoYun Cheng

  2. Department of Mathematics, Nanjing University, Nanjing, 210093, China

    XueJun Guo & HouRong Qin

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  1. XiaoYun Cheng
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  2. XueJun Guo
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  3. HouRong Qin
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Correspondence to XiaoYun Cheng.

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Cheng, X., Guo, X. & Qin, H. The densities for 3-ranks of tame kernels of cyclic cubic number fields. Sci. China Math. 57, 43–47 (2014). https://doi.org/10.1007/s11425-013-4622-0

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