On \(\ell\)-torsion in class groups of number fields
Let \(K\) be an algebraic number field of degree \(d\) over \(\mathbb Q\), \(D_K = |\text{disc}\,K/\mathbb Q|\) the absolute value of the discriminant of \(K\), \(\ell\) a natural number and \(\text{Cl}_K[\ell]\) the \(\ell\)-torsion subgroup of the class group of \(K\). Put \(\delta = \frac 1{2\ell (d-1)}\). Assuming GRH, the first author and \textit{A. Venkatesh} [Int. Math. Res. Not. 2007, No. 1, Article ID rnm002, 18 p. (2007; Zbl 1130.11060)] showed that for any \(\varepsilon >0\) one has \[ |\text{Cl}_K[\ell]| \ll_{d, \ell, \varepsilon} D_K^{\frac 12 - \delta + \varepsilon}. \] In the present paper, the authors prove that this result holds without assuming GRH for \(d \in \{2,3,4,5\}\) for all fields \(K\) with \(D_K \leq X\) up to some small set of exceptions (which is of density \(0\)). For \(d=4\) one must suppose that the Galois group of the normal closure of \(K\) is not \(D_4\) and that \(\ell \geq8\); and for \(d=5\) that \(\ell \geq 25\). From this result one can deduce that under the same suppositions one has \[ \sum_{\overset{\deg(K)=4}{0 < D_K \leq X}} |\text{Cl}_K[\ell]| \ll_{d,\varepsilon} D_K^{\frac {3}{2} - \delta + \varepsilon}. \] Following the ideas of Ellenberg and Venkatesh [loc. cit.], the exceptions are those fields \(K\) which have only ``few'' ``small'' primes which split completely. Therefore the authors develop a new sieve, called ``Chebyshev'' sieve (Section 3 of the paper), and asymptotically count the number of the fields under consideration with \(D_K \leq X\), using results which go back to \textit{M. Bhargava} [Ann. Math. (2) 162, No. 2, 1031--1063 (2005; Zbl 1159.11045)] and [Ann. Math. (2) 172, No. 3, 1559--1591 (2010; Zbl 1220.11139)].
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