On the μ-invariant of Katz p-adic L-functions attached to imaginary quadratic fields

Abstract

We extend to p = 2 and p = 3 the result of Gillard and Schneps which says that the µ-invariant of Katz p-adic L functions attached to imaginary quadratic fields is zero for p > 3. The arithmetic interpretation of this fact is given in the introduction.

On Tamely Ramified Iwasawa Modules for $\Zp$-extensions of Imaginary Quadratic Fields

“…We see that μ(K c /k) = 0 by Ferrero-Washington's theorem [6]. Gillard [10], [11], Schneps [26] (and recently Oukhaba-Viguié [20]) showed μ = 0 for certain non-cyclotomic Z p -extensions. Bloom-Gerth [1] gave an upper bound of the number of Z p -extensions satisfying μ > 0 for a fixed k (see Section 3.2).…”

Weak Greenberg's generalized conjecture for imaginary quadratic fields

“…On the Z p -extension N ∞ /k, it is known that the Iwasawa µ-invariant of N ∞ /k is zero. This was proved by Gillard [10] and Schneps [30] for p ≥ 5 and Oukhaba-Viguié [27] for p = 2, 3. By [21, Proposition 1.C], the Iwasawa λ-invariant of N ∞ /k is zero if and only if every ideal class of the p-part of the ideal class group of k becomes principal in N ∞ .…”

“…∞,i is isomorphic to Z p for each i. From [10,30,27], we obtain µ(N ∞ /k) = 0. Furthermore, since we suppose that λ(N ∞ /k) = 0, Gal(L N∞ /N ∞ ) is finite.…”

“…Hence we have I P * ∞,i ∼ = Z p . Using [10,30,27], we obtain µ(H ∞ /H) = 0 since H/k is an abelian extension. This implies that Gal(M p * (H ∞ )/H ∞ ) is a finitely generated as a Z p -module.…”

Analogues of Iwasawa’s $\mu = 0$ conjecture and the weak Leopoldt conjecture for a non-cyclotomic $\mathbb{Z}_2$-extension

“…We stress that this result was unknown previously, except in the very special case when E is the quadratic twist of A by the compositum with H of a quadratic extension of K. Moreover, none of the analytic results is known for such a curve E, for example, the construction of the p-adic L-function attached to E. Our proof of Theorem 1.1 uses an elliptic analogue of Sinnott's beautiful proof of the vanishing of the cyclotomic µ-invariant. Considerable past work in this direction has already been done by Gillard [12], [11] and Schneps [17] for split odd primes p. For the prime p = 2 there has recently been independent work by Oukhaba and Viguié [18], which would seemingly include a proof of Theorem 1.1. However, we give the full details of a rather different construction of the p-adic L-functions and the analogue of Sinnott's proof in our case, rather than the arguments sketched in [18].…”