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Title: A modular construction of unramified 𝑝-extensions of ℚ(ℕ^{1/𝕡})

We show that for primesN,p≥<#comment/>5N, p \geq 5withN≡<#comment/>−<#comment/>1modpN \equiv -1 \bmod p, the class number ofQ(N1/p)\mathbb {Q}(N^{1/p})is divisible bypp. Our methods are via congruences between Eisenstein series and cusp forms. In particular, we show that whenN≡<#comment/>−<#comment/>1modpN \equiv -1 \bmod p, there is always a cusp form of weight22and levelΓ<#comment/>0(N2)\Gamma _0(N^2)whoseℓ<#comment/>\ellth Fourier coefficient is congruent toℓ<#comment/>+1\ell + 1modulo a prime abovepp, for all primesℓ<#comment/>\ell. We use the Galois representation of such a cusp form to explicitly construct an unramified degree-ppextension ofQ(N1/p)\mathbb {Q}(N^{1/p}).

 
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Award ID(s):
1901867
PAR ID:
10473940
Author(s) / Creator(s):
;
Publisher / Repository:
American Mathematical Society
Date Published:
Journal Name:
Proceedings of the American Mathematical Society, Series B
Volume:
9
Issue:
39
ISSN:
2330-1511
Page Range / eLocation ID:
415 to 431
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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