The Iwasawa λ-invariants of ℤₚ-extensions of real quadratic fields
1. Introduction. Let k be a totally real number field. Let p be a fixed prime number and ℤₚ the ring of all p-adic integers. We denote by λ=λₚ(k), μ=μₚ(k) and ν=νₚ(k) the Iwasawa invariants of the cyclotomic ℤₚ-extension $k_∞$ of k for p (cf. ). Then Greenberg's conjecture states that both λₚ(k) and μₚ(k) always vanish (cf. ). In other words, the order of the p-primary part of the ideal class group of kₙ remains bounded as n tends to infinity, where kₙ is the nth layer of $k_∞/k$. We know by the Ferrero-Washington theorem (cf. , ) that μₚ(k) always vanishes when k is an abelian (not necessarily totally real) number field. However, the conjecture remains unsolved up to now except for some special cases (cf. , , -, ). This paper is a continuation of our previous papers , - and , that is to say, we investigate Greenberg's conjecture when k is a real quadratic field and p is an odd prime number which splits in k. The purpose of this paper is to extend our previous results, and to give basic numerical data of k=ℚ(√m) for 0 ≤ m ≤ 10000 and p=3. On the basis of these data, we can verify Greenberg's conjecture for most of these k's.