Triviality of the ℓ-class groups in $\mathbb{Z}_{p}$-extensions of $\mathbb{Q}(\sqrt{-1})$ for split primes p ≡ 1 modulo 4.

In this paper we study the class numbers in the finite layers of certain non-cyclotomic $\mathbb{Z}$p-extensions of the imaginary quadratic field $\mathbb{Q}(\sqrt{-1})$, for all primes p ≡ 1 modulo 4. By studying the Mahler measure of elliptic units, we are able to show that there are only finitely many primes ℓ congruent to a primitive root modulo p2 that divide any of the class numbers in the $\mathbb{Z}$p-extension. [ABSTRACT FROM AUTHOR]