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Reflection principles and bounds for class group torsion. (English) Zbl 1130.11060

For an odd prime \(\ell\), the authors give some upper bounds on the order of the \(\ell\)-class group of a number field. The notation \(A\ll_B C\) is defined to mean that there is a function \(f(B)\) such that \(A\leq f(B)C\). The main results are: Let \([K: \mathbb{Q}]= d\). Then assuming the GRH \[ \#\text{Cl}_K(\ell)\ll_{d,\varepsilon}(\text{disc }K)^{{1\over 2}-{1\over 2\ell(d-1)}+\varepsilon}, \] where \(\text{Cl}_K(\ell)\) and disc \(K\) denote the \(\ell\)-class group and discriminant of \(K\), respectively. Unconditionally, the authors prove: If \(D\) is a square-free integer and \(K= \mathbb{Q}(\sqrt{D})\) then \(\text{Cl}_K(3)\ll_\varepsilon|D|^{{1\over 3}+\varepsilon}\). Also, let \(\zeta\) denote a primitive \(\ell\)th root of unity and \(K_0= \mathbb{Q}(\zeta+ \zeta^{-1})\) and \([K:K_0]= d\) be even such that \(\zeta\not\in K\) and \(K(\zeta)/K_0\) has no intermediary fields except \(K\) and \(\mathbb{Q}(\zeta)\). Then \[ \#\text{Cl}_K(\ell)\ll_{\varepsilon, d,\ell}(\text{dsc }K)^{{1\over 2}- {1\over 2\ell d(d- 1)}+\varepsilon}. \] The proofs of the results combine the use of noninert primes, an Arakelov version of the class group and reflection principles of Scholz type.

MSC:

11R29 Class numbers, class groups, discriminants
11R11 Quadratic extensions
11R20 Other abelian and metabelian extensions
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