Kida’s formula for split prime ℤ p -extensions over imaginary quadratic fields

Let p be an odd prime number and k an imaginary quadratic field in which p splits into 𝔭 and 𝔭 * . Then there exists a uniquely defined ℤ p -extension N ∞ / k such that the prime ideal 𝔭 * does not ramify. For a finite extension K / k , we call K ∞ = K N ∞ the split prime ℤ p -extension corresponding to 𝔭 . We prove an analogue of Kida’s formula for the split prime ℤ p -extensions. As an application, we apply this formula to 𝔭 -ramified Iwasawa modules and determine the isomorphism classes of unramified Iwasawa modules associated to ℤ p -extensions over k .