A modular construction of unramified 𝑝-extensions of ℚ(ℕ^{1/𝕡})

We show that for primes N , p ≥ 5 N, p \geq 5 with N ≡ − 1 mod p N \equiv -1 \bmod p , the class number of Q ( N 1 / p ) \mathbb {Q}(N^{1/p}) is divisible by p p . Our methods are via congruences between Eisenstein series and cusp forms. In particular, we show that when N ≡ − 1 mod p N \equiv -1 \bmod p , there is always a cusp form of weight 2 2 and level Γ 0 ( N 2 ) \Gamma _0(N^2) whose ℓ \ell th Fourier coefficient is congruent to ℓ + 1 \ell + 1 modulo a prime above p p , for all primes ℓ \ell . We use the Galois representation of such a cusp form to explicitly construct an unramified degree- p p extension of Q ( N 1 / p ) \mathbb {Q}(N^{1/p}) .