Let K be a number field and $p$ an odd prime. Let $\mu$ be the Iwasawa $\mu$-invariant of the class group of the cyclotomic $\mathbb{Z}_p$-extension of $K$. If $K$ is abelian over $\mathbb{Q}$ then it is known that $\mu=0$ (Ferrero-Washinton, see Washington 7.5). Iwasawa conjectured that $\mu=0$ for all $K$.
Is something known for the case when $K$ is abelian over an imaginary quadratic field $k$ ?
Best Answer
If I am not mistaken, it proves that mu invariant of the Z_p times Z_p extension is 0 and this was Schneps's thesis. It is unfortunately not enough to show the conjecture of Iwasawa in this case even using the vanishing of anticyclotomic mu invariant proven by Hida.
Edit: In fact, what Schneps proves is that the mu invariant of the Z_p extension in which only one of the primes above p is ramified.