Stickelberger Ideals and Relative Class Numbers in Function Fields

✍ Scribed by Linsheng Yin

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 139 KB
• Volume: 81
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1999.2472

✦ Synopsis

It is well-known that in the cyclotomic number field Q(pn) of prime power conductor, where pn=exp(2?i p n ), the index of cyclotomic units in the total unit group is equal to the class number of its maximal real subfield, which is due to Kummer. On the other hand, in 1962 Iwasawa [Iw] showed that the index of the Stickelberger ideal associated to the field Q(`pn) is equal to the relative class number of the field. Sinnott [Si] extended these results to a general cyclotomic field in 1978 by introducing the powerful device of cohomology to the computation of the indices. The analogue of Kummer Sinnott's unit-index calculation, with the Carlitz module assigned to the role played in classical cyclotomic theory by the multiplicative group, was carried out by Galovich and Rosen [GR]. The author [Yi1-2] generalized this unit-index by replacing the Carlitz module with a general sign-normalized rank one Drinfeld module conditional on a conjecture concerning the Galois-module structure of the sign-cohomology of the universal ordinary distribution associated to a global function field. Recently Anderson [An] invented a remarkable method for computing the sign-cohomology and decided its structure completely. In this paper, using the structure of the sign-cohomology we give the analogue and the generalization of Iwasawa Sinnott's [Iw, Si] index formula of Stickelberger ideal in the theory of sign-normalized rank one Drinfeld modules.