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Averages and moments associated to class numbers of imaginary quadratic fields. (English) Zbl 1391.11151

For any odd prime number \(3 \leq \ell \in \mathbb P\) and any square-free integer \(d \in \mathbb N\) let \(h_\ell(-d)\) denote the \(\ell\)-part of the class number of the number field \(\mathbb Q(\sqrt{-d})\). Motivated by the Cohen-Lenstra heuristics, the authors give upper bounds for the averages and higher moments of \(h_\ell(-d)\), \[ \sum_{0 < d < X} h_\ell(-d)^k, \] where \(1 \leq k\) and the sum runs over the square-free integers \(d < X\). Up to the cases \(\ell =3\) or \(5\) and \(k=1\), these seem to be the first non-trivial such upper bounds. Using Scholz’s reflection principle, the results can also be transferred to the averages and moments of \(h_3(d)\), the \(3\)-part of the class number of the number field \(\mathbb Q(\sqrt{d})\).
The key ingredients for the proof are the technical Proposition 2.1, which rests on a remark of K. Soundararajan [J. Lond. Math. Soc., II. Ser. 61, No. 3, 681–690 (2000; Zbl 1018.11054)], giving an upper bound for \(h_\ell(-d)\) up to some controllable set of exceptions, as well as the technical Propositions 2.3 and 2.4.

MSC:

11R29 Class numbers, class groups, discriminants
11D45 Counting solutions of Diophantine equations

Citations:

Zbl 1018.11054
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References:

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