Abstract: In 1922, Nagell proved that for any integer m, there exist infinitely many imaginary quadratic number fields with class number divisible by m. Over a half-century later, in 1984, Azuhata and Ichimura generalized Nagell's result to number fields of arbitrary degree. More generally, they proved, for any integers m,n>1, the existence of infinitely many number fields of degree n with ideal class group of m-rank at least [n/2] (Nakano later improved this to [n/2]+1). Recently, progress has also been made on enumerating the number fields in such results. Using an apparently new technique, we improve on the Azuhata-Ichimura-Nakano result and others, including enumerative results. Our technique relies on the Hilbert Irreducibility Theorem and finding certain curves whose Jacobians have a large rational torsion subgroup. The material will be presented in a fairly elementary, accessible manner (for example, largely avoiding any discussion of Jacobians).
This talk will be accessible to graduate students.