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

Abstract

Let F be a cubic cyclic field with t (⩾ 2) ramified primes. For a finite abelian group G, let r3(G) be the 3-rank of G. If 3 does not ramify in F, then it is proved that t−1 ⩽ r3(K2OF) ⩽ 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 r3(K2OF) = 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.