Class numbers of real cyclotomic fields
by
Leanne Robertson
Smith College
Coauthors: Joe Buhler (Reed College and CCR), Carl Pomerance (Dartmouth College)

Let h⁺(pⁿ) denote the class number of the maximal totally real subfield Q(cos(2\pi/pⁿ)) of the field of pⁿ-th roots of unity. We show that (speculative extensions of) the Cohen-Lenstra heuristics on class groups provide support for the following conjecture: for all but finitely many pairs (p, n), where p is a prime and n is a positive integer, h⁺(pⁿ⁺¹) = h⁺(pⁿ). In particular, this predicts that for all but finitely many primes p, h⁺(pⁿ) = h⁺(p ) for all positive integers n. We also discuss work in progress to test this heuristic prediction empirically by searching for Jordan-Hölder factors of prime-power order q of these class groups for ``small'' p, n, and q (this follows Schoof’s work on the class numbers h⁺(p)). This is joint work with Joe Buhler and Carl Pomerance.

Date received: April 30, 2004