The Gauss Higher Relative Class Number Problem

Assuming the 2-adic Iwasawa main conjecture, we find all CM fields with higher relative class number at most 16: there are at least 31 and at most 34 such fields, and exactly one is not abelian. The problem of determining all imaginary quadratic fields K = Q( √ d) of small class number h(K) was first posed in Article 303 of Gauss’ Disquisitiones Arithmeticae. It would take almost 150 years of work, culminating in the results of Stark [19] and Baker [1], to determine those fields with class number at most two: there are exactly 27, the last having discriminant d = −427. (See Goldfeld [6] or Stark [21] for a history of this problem.) Significant further progress has been made recently by Watkins [24], who enumerated all such fields K with class number h(K) ≤ 100. One interesting generalization of the Gauss class number problem is to replace Q by a totally real field F . Let K/F be a CM extension, i.e., K is a totally imaginary quadratic extension of a totally real field F , and let [F : Q] = n. We have the divisibility relation h(F ) | h(K), and we denote by h(K) = h(K)/h(F ) the relative class number. It is known that there are only finitely many CM fields with fixed relative class number [20]. The complete list of CM fields of relative class number one is still unknown (see Lee-Kwon [11] for an overview). However, many partial results are known: for example, there are exactly 302 imaginary abelian number fields K with relative class number one [3], each having degree [K : Q] ≤ 24. The integer h(K) can be determined by the analytic relative class number formula, as follows. Let χ : Gal(K/F ) → {±1} denote the nontrivial character associated to the extension K/F and let L(χ, s) denote the Artin L-function associated to χ. Then L(χ, 0) = 2 Q(K) h(K) w(K) , where w(K) = #μ(K) is the number of roots of unity in the field K and Q(K) = [Z∗K : Z ∗ F μ(K)] ∈ {1, 2} is the Hasse Q-unit index. In this article, we consider the further generalization of the Gauss problem to higher relative class numbers of CM fields. Let E be a number field with ring of integers ZE and let m ∈ Z≥3 be an odd integer. We define the higher class group of E to be H(SpecZE ,Z(m)) = ∏ p H ét(SpecZE [1/p],Zp(m)). The group H(SpecZE ,Z(m)) is finite, and we let hm(E) denote its order. For p 6= 2, the Quillen-Lichtenbaum conjecture (which appears to have been proven by Voevodsky-Rust-Suslin-Weibel) implies that the étale l-adic Chern character K2m−2(ZE)⊗ Zp ∼ −→ H ét(SpecZE [1/p],Zp(m)) 1