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

In his 1976 proof of the converse of Herbrand’s theorem, Ribet used Eisenstein-cuspidal congruences to produce unramified degree-p extensions of the p-th cyclotomic field when p is an odd prime. After reviewing Ribet’s strategy, we will discuss recent work with Preston Wake in which we apply similar techniques to produce unramified degree-p extensions of Q(N1/p) when N is a prime that is congruent to -1 mod p. This answers a question posed on Frank Calegari’s blog.