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  1. ; ; (, Research in Number Theory)
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  2. ; ; ; ; (, Forum of Mathematics, Sigma)
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  3. ; (, Proceedings of the American Mathematical Society, Series B)
    We show that for primes N , p ≥<#comment/> 5 N, p \geq 5 with N ≡<#comment/> −<#comment/> 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 ≡<#comment/> −<#comment/> 1 mod p N \equiv -1 \bmod p , there is always a cusp form of weight 2 2 and level Γ<#comment/> 0 ( N 2 ) \Gamma _0(N^2) whose ℓ<#comment/> \ell th Fourier coefficient is congruent to ℓ<#comment/> + 1 \ell + 1 modulo a prime above p p , for all primes ℓ<#comment/> \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}) . 
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  4. ; (, Proceedings of the National Academy of Sciences)
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  5. (, Journal of the European Mathematical Society)
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  6. ; (, Advances in Mathematics)
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  7. ; (, Duke Mathematical Journal)
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