Weber’s Class Number One Problem

Class numbers in cyclotomic Zp-extensions

“…Fukuda, Komatsu and Morisawa proved a similar result for the Z 3 -extension [9], which suggests that all the B 3,n too may have trivial class group.…”

“…This conjecture is further supported by the work of Morisawa, who proved in his thesis [16] that all primes p less than 400,000 do not the class number of B 3,n for all n. Morisawa's result has recently been further improved by Fukuda, Komatsu and Morisawa [9], who proved that no prime less than 10 9 divides the class number of B 3,n .…”

Tate–Shafarevich groups in the cyclotomic Zˆ-extension and Weber's class number problem

“…), e ≥ 1, whose p-rank is a multiple of the residue degree ρ N of p in Q p (µ N )/Q p ; thus ρ N → ∞ as N → ∞, which is considered "incredible" for arithmetic invariants, as class groups, for totally real fields. Indeed, interesting examples occur more easily when p totally splits in Q(µ N ) (i.e., p ≡ 1 (mod N )) and this "explains" the result of [38] and [39] claiming that # C Q(ℓ n ) is odd in Q(ℓ ∞ ) for all ℓ < 500, that of [37,51,52] and explicit deep analytic computations in [5,10,11,14,36,37,38,39,48,49,51,52,59] (e.g., Washington's theorem [59] claiming that for ℓ and p fixed, # C K is constant for all n large enough, whence C * K = 1 for all n ≫ 0, then [14, Theorems 2, 3, 4, Corollary 1]); mention also the numerous pioneering Horie's papers proving results of the form: "let ℓ 0 be a small prime; then a prime p, totally inert in some Q(ℓ n 0 0 ), yields C Q(ℓ n 0 ) = 1 for all n". In [5], a conjecture (from "speculative extensions of the Cohen-Lenstra-Martinet heuristics") implies C * Q(ℓ n ) = 1 for finitely many layers (possibly none).…”

“…Indeed, one may ask if the arithmetic of these fields is as smooth as it is conjectured (for the class group C Q(N ) ) by many authors after many verifications and partial proofs [2,5,10,11,12,13,14,15,34,35,36,37,38,39,40,46,47,48,49,50,51,52,59]. The triviality of C Q(ℓ n ) has, so far, no counterexamples as ℓ, n, p vary, but that of the Tate-Shafarevich group T Q(ℓ n ) (or more generally T Q(N ) ) is, on the contrary, not true as we shall see numerically, and, for composite N , few C Q(N ) = 1 have been discovered.…”

Class Numbers of Quadratic Fields

“…In [Web1886], Weber showed that h 2,n is odd for all n ≥ 1. Subsequently, Iwasawa [Iwa56] generalized Weber's result to show that for all n ≥ 1, the class number h p,n is not divisible by p. Then, in [FKM14], Fukuda and Komatsu proved that h 2,n is not divisible by any prime less than 10 9 . This led to the following conjecture, often called Weber's class number problem: Conjecture 1.…”