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

Abstract

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

12 pages