Ellenberg, Jordan S.; Venkatesh, Akshay Reflection principles and bounds for class group torsion. (English) Zbl 1130.11060 Int. Math. Res. Not. 2007, No. 1, Article ID rnm002, 18 p. (2007). 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. Reviewer: Charles Parry (Blacksburg) Cited in 6 ReviewsCited in 33 Documents MSC: 11R29 Class numbers, class groups, discriminants 11R11 Quadratic extensions 11R20 Other abelian and metabelian extensions Keywords:\(\ell\)-class group PDFBibTeX XMLCite \textit{J. S. Ellenberg} and \textit{A. Venkatesh}, Int. Math. Res. Not. 2007, No. 1, Article ID rnm002, 18 p. (2007; Zbl 1130.11060) Full Text: DOI Link