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Journal article
Date
2022-10-31
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Mathematics
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DOI
http://dx.doi.org/10.1090/bproc/141
Abstract
We show that for primes $N, p \geq 5$ with $N \equiv -1 \bmod p$, 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 $N \equiv -1 \bmod p$, 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 $\mathbb {Q}(N^{1/p})$.
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Proceedings of the American Mathematical Society, Series B, Vol. 9
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