Onp-adicL-functions and Zp-extensions of Certain Real Abelian Number Fields

✍ Scribed by Hisao Taya

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 156 KB
• Volume: 75
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1998.2326

✦ Synopsis

Let k be a real abelian number field with Galois group 2 and p an odd prime number. Denote by k the cyclotomic Z p -extension of k with Galois group 1 and by k n the nth layer of k Âk. Assume that the order of 2 is prime to p and that p splits completely in kÂQ. In this article, we describe the order of the 1-invariant part of the 9-component of the p-Sylow subgroup of the ideal class group of k n for sufficiently large n, in terms of a special value of the p-adic L-function associated to 9, where 9 is an irreducible Q p -character of 2. This allows us to obtain an alternative formulation of Greenberg's criterion for the vanishing of the 9-components of the cyclotomic Iwasawa *-and +-invariants of k for p. We also compute some examples for cyclic cubic fields and p=5, 7.