On the 4-rank of class groups of quadratic number fields
The Cohen-Lenstra conjectures deal with densities of quadratic number fields whose class groups have given \(p\)-rank. In order to state them, let \(D\) denote the discriminant of a quadratic number field, \(C_D\) its class group, and \(C^+_D\) its narrow class group. For an abelian group \(A\), \(r_p(A)\) denotes the \(p\)-rank of \(A\), and \(r_4(A) = r_2(A^2)\) its \(4\)-rank. For a numerical function \(f(D)\) defined on the set of positive discriminants, we say that \(f\) has a mean value \({\mathcal M}^+(f)\) over positive discriminants if the quotient \(\frac{\sum_{0 < D < X} f(D)}{\sum_{0 < D < X} 1}\) has limit \({\mathcal M}^+(f)\) as \(X \to \infty\). The corresponding quantity for negative discriminants is denoted by \({\mathcal M}^-(f)\). Two of the main conjectures of \textit{H. Cohen} and \textit{H. W. Lenstra jun.} [Lect. Notes Math. 1068, 33--62 (1984; Zbl 0558.12002), see also Lect. Notes Math. 1052, 26--36 (1984; Zbl 0532.12008)] (slightly modified by \textit{F. Gerth} [Expo. Math. 5, 181--184 (1987; Zbl 0613.12003)]) can be formulated as follows: for any prime \(p\) and every integer \(\alpha \geq 0\) we have \(C^-(p,\alpha): {\mathcal M}^- (\prod_{0 \leq i < \alpha} (p^{r_k(C_D^2)} - p^i))= 1\) and \(C^+(p,\alpha): {\mathcal M}^+ (\prod_{0 \leq i < \alpha} (p^{r_k(C_D^2)} - p^i))= p^{-\alpha}\). Let \(\eta_k(t) = \prod_{j=1}^k (1-t^{-j})\) for integers \(k \geq 0\) or \(k= \infty\). Then the authors have shown in [Algorithmic number theory. 7th international symposium, ANTS-VII, Lect. Notes Comput. Sci. 4076, 40--55 (2006; Zbl 1143.11352)] that the truth of the conjectures \(C^\pm(p,\alpha)\) for \(p\) and all integers \(\alpha \geq 0\) implies the following assertions, which are also part of the Cohen-Lenstra conjectures extended by Gerth: 1. The density of negative fundamental discriminants \(D\) with \(r_k(C_D^2) = r\) is equal to \[ p^{-r^2} \eta_\infty(p) \eta_r(p)^{-2}. \] 2. The density of positive fundamental discriminants \(D\) with \(r_k(C_D^2) = r\) is equal to \[ p^{-r(r+1)} \eta_\infty(p) \eta_r(p)^{-1} \eta_{r+1}(p)^{-1}. \] The conjectures \(C^\pm(p,0)\) are trivially true, and \(C^\pm(p,1)\) is known to hold for \(p = 3\) as a consequence of results of Davenport and Heilbronn. In this article, the authors prove Gerth's conjectures \(C^\pm(2,\alpha)\) for every integer \(\alpha \geq 0\); this then implies the conjectures 1 and 2. The proof itself is, of course, rather involved. It begins with rewriting Rédei's formula for the \(4\)-rank of quadratic fields using the quadratic Hilbert symbol; other ingredients include an idea of \textit{D. R. Heath-Brown} [Invent. Math. 118, 331--370 (1994; Zbl 0815.11032)] used for computing the average \(2\)-rank of the Selmer groups of the elliptic curves coming from the congruent number problem. The massive combinatorial problems involved are solved by counting subspaces in a suitably defined finite geometry.
scientific article; zbMATH DE number 5145033