A modular construction of unramified p-extensions of ℚ (N1/p)

Abstract

We show that for primes N, p ≥ 5 with N ≡ −1 mod p, the class number of ℚ(N1/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 Γ0(N2) whose ℓth 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(N1/p).