| Heuristics for class numbers of prime-power real cyclotomic fields (2008) | |||||||||||||
Abstract | |||||||||||||
| Dedicated to Hugh Williams on the occasion of his sixtieth birthday Abstract. Let h + (ℓ n) denote the class number of the maximal totally real subfield Q(cos(2π/ℓ n)) of the field of ℓ n-th roots of unity. The goal of this paper is to show that (speculative extensions of) the Cohen-Lenstra heuristics on class groups provide support for the following conjecture: for all but finitely many pairs (ℓ, n), where ℓ is a prime and n is a positive integer, h + (ℓ n+1) = h + (ℓ n). In particular, this predicts that for all but finitely many primes ℓ, h + (ℓ n) = h + (ℓ) for all positive integers n. It is possible that there are no exceptional primes ℓ at all. Extensive computations of René Schoof [9] enumerate all “small ” components of the plus part of the class group of cyclotomic fields of prime conductor. Let ℓ be an odd prime, K(ℓ): = Q(ζℓ + ζ −1 ℓ) be the maximal totally real subfield of ℓ-th roots of unity, and G(ℓ): = Gal (K(ℓ)/Q) be the Galois group of K(ℓ) over Q, so that G(ℓ) is a cyclic group of order (ℓ − 1)/2. The class group Cl + (ℓ) of K(ℓ) is a module over the group ring Z[G(ℓ)]. For all ℓ < 10000, Schoof finds the largest subgroup of the class group whose simple factors (as Z[G(ℓ)] modules) have size less than 80000. Let h + (ℓ) denote the order of the class group of K(ℓ), and ˜ h + (ℓ) denote the order of Schoof’s subgroup. For all ℓ < 10000 either h + (ℓ) = ˜ h + (ℓ) or h + (ℓ)> 80000 ˜h + (ℓ); it seems very likely that h + (ℓ) = ˜ h + (ℓ) in every case. In fact, the largest simple factor found in the search has order 1451, there are 2 others over 500, and almost all of the others are below 100. The novelty and extent of these computations are indicated by the fact that h + (ℓ) is known only for ℓ ≤ 67 (or ℓ ≤ 163 assuming the GRH); the exact computations of h + (ℓ) rely on bounds on 1 c○0000 American Mathematical Society | |||||||||||||
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