On the p-adic Leopoldt transform of a power series
Let be an imaginary cyclic quartic number field whose 2-class group is of type , i.e., isomorphic to . The aim of this paper is to determine the structure of the Iwasawa module of the genus field of .
1. Introduction. Let p be a prime number and the ring of p-adic integers. Let k be a finite extension of the rational number field ℚ, a -extension of k, the nth layer of , and the p-Sylow subgroup of the ideal class group of . Iwasawa proved the following well-known theorem about the order of : Theorem A (Iwasawa). Let be a -extension and the p-Sylow subgroup of the ideal class group of , where is the th layer of . Then there exist integers , , , and n₀ ≥ 0 such that for...
The purpose of this work is to carry out the first step in our four-step program towards the main conjecture for by the method of Eisenstein congruence on , where is an imaginary quadratic field. We construct a -adic family of ordinary Eisenstein series on the group of unitary similitudes with the optimal constant term which is basically the product of the Kubota-Leopodlt -adic -function and a -adic -function for . This construction also provides a different point of view of -adic...