摘要
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(K2O 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 r3(K2O 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∞,r=3-r2∞k=1(1-3-k)r k=1(1-3-k)2.This suggests that the Cohen-Lenstra conjecture for ideal class groups can be extended to the tame kernels of cyclic cubic number fields.
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（K2O 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 r3（K2O 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∞,r=3-r2∞k=1（1-3-k）r k=1（1-3-k）2.This suggests that the Cohen-Lenstra conjecture for ideal class groups can be extended to the tame kernels of cyclic cubic number fields.
基金
supported by National Natural Science Foundation of China (Grant Nos. 11201225,11271177,10971091 and 11171141)
Natural Science Foundation of the Jiangsu Province (Grant Nos. BK2010007 and BK2010362)
Program for New Century Excellent Talents in University (Grant No. NCET-100471)
