Ideal class groups and torsion in Picard groups of varieties

Details Ideal class groups and torsion in Picard groups of varieties Ideal class groups and torsion in Picard groups of varieties

2008-05-09

arxiv.org/abs/0805.1361v1

Mathematics

Number Theory

31 pages

Scientific paper

We give a new general technique for constructing and counting number fields with an ideal class group of nontrivial m-rank. Our results can be viewed as providing a way of specializing the Picard group of a variety V over $\mathbb{Q}$ to obtain class groups for number fields $\mathbb{Q}(P)$, $P\in V(\Qbar)$, for certain families of points P. In particular, we show how the problem of constructing quadratic number fields with a large-rank ideal class group can be reduced to the problem of finding a hyperelliptic curve with a rational Weierstrass point and a large rational torsion subgroup in its Jacobian. Furthermore, we show how many previous results on constructing large-rank ideal class groups can be fit into our framework and rederived. As an application of our technique, we derive a quantitative version of a theorem of Nakano. This gives the best known general quantitative result on number fields with a large-rank ideal class group.

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