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

Abstract

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