Ideal Class Groups and Rational Torsion in Jacobians of Curves
<BR><BR>

Aaron Levin <BR><BR>

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.