We use Galois cohomology to study the p-rank of the class group of Q(N 1/p ), where N ≡ 1 mod p is prime. We prove a partial converse to a theorem of Calegari-Emerton, and provide a new explanation of the known counterexamples to the full converse of their result. In the case p = 5, we prove a complete characterization of the 5-rank of the class group of Q(N 1/5 ) in terms of whether or not (N −1)/2 k=1 k k and √ 5−1 2 are 5th powers mod N .

We develop a new strategy for studying low weight specializations of p-adic families of ordinary modular forms. In the elliptic case, we give a new proof of a result of Ghate-Vatsal which states that a Hida family contains infinitely many classical eigenforms of weight one if and only if it has complex multiplication. Our strategy is designed to explicitly avoid use of the related facts that the Galois representation attached to a classical weight one eigenform has finite image, and that classical weight one eigenforms satisfy the Ramanujan conjecture. We indicate how this strategy might be used to prove similar statement in the case of partial weight one Hilbert modular forms, given a suitable development of Hida theory in that setting.

We determine the structure of the quotient of the free group on 26 generators by English language anagrams. This group admits a surprisingly simple presentation as a quotient of the free group by 301 of the possible 325 commutators of pairs of generators; all of the 24 missing commutators involve at least one of the letters j, q, x, z. We describe the algorithm which can be used to determine this group given any dictionary, and provide examples from the SOWPODS scrabble dictionary witnessing the 301 commutators found.

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