Ideal Class Groups and Rational Torsion in Jacobians of Curves

Aaron Levin

We study the problem of constructing and enumerating, for any integers m, n> 1, number fields of degree n whose ideal class groups have "large" m-rank. Our technique, which appears to be new, relies on the Hilbert Irreducibility Theorem and finding certain curves whose Jacobians have a large rational torsion subgroup. Using this technique we improve on results of Nakano, Bilu-Luca, and others.