O ct 2 00 8 Kummer Generators and Lambda Invariants

Let F0 = Q(√ −d) be an imaginary quadratic field with 3 ∤ d and let K0 = Q(√ 3d). Let ε0 be the fundamental unit of K0 and let λ be the Iwasawa λ-invariant for the cyclotomic Z3-extension of F0. The theory of 3-adic L-functions gives conditions for λ ≥ 2 in terms of ǫ0 and the class numbers of F0 and K0. We construct units of K1, the first level of the Z3-extension of K0, that potentially occur as Kummer generators of unramified extensions of F1(ζ3) and which give an algebraic interpretation of the condition that λ ≥ 2. We also discuss similar results on λ ≥ 2 that arise from work of Gross-Koblitz. Let F 0 = Q(√ −d) be an imaginary quadratic field and let h − be its class number. Let ε 0 and h + be the fundamental unit and class number of K 0 = Q(√ 3d). The starting point of this work is the following observation. If 3 splits in F 0 , and h − , h + , and ε 0 satisfy certain congruences, then the theory of 3-adic L-functions shows that the Iwasawa λ-invariant for the cyclotomic Z 3-extension F ∞ /F 0 satisfies λ ≥ 2. This means that, as we go up the tower of fields in the Z 3-extension, eventually the 3-Sylow subgroup of the ideal class group has rank at least 2. Therefore, there are at least two independent unramified abelian extensions of degree 3. We show in Section 7 that, after we adjoin a cube root of unity, ε 0 gives a Kummer generator for one of these unramified extensions. The question is then, where is the other Kummer generator? Since a condition on ε 0 is used to show this generator exists, the goal is to relate this generator to ε 0. This we do in Sections 6 and 8. In fact, inspired by ideas from genus theory, in Section 6 we find special generators for the part of the units where the potential Kummer generators live. The congruence condition on ε 0 can then be interpreted in terms of these generators. This gives an algebraic interpretation of the condition for λ ≥ 2. In Section 9, we use a result of Gross-Koblitz to obtain similar results. There is a difference in this approach: In Section 9, we use the value of the …