Weber’s class number problem in the cyclotomic ℤ₂-extension of ℚ, II

Let h_n denote the class number of n-th layer of the cyclotomic ℤ₂-extension of ℚ. Weber proved that h_n (n ≥ 1) is odd and Horie proved that h_n (n ≥ 1) is not divisible by a prime number ℓ satisfying ℓ ≡ 3, 5 (mod 8). In a previous paper, the authors showed that h_n (n ≥ 1) is not divisible by a prime number ℓ less than 10⁷. In this paper, by investigating properties of a special unit more precisely, we show that h_n (n ≥ 1) is not divisible by a prime number ℓ less than 1.2 • 10⁸. Our argument also leads to the conclusion that h_n (n ≥ 1) is not divisible by a prime number ℓ satisfying ℓ = ± 1 (mod 16).