A hyperelliptic function field is a field of the form k(x,y) where k is a finite field of odd characteristic and y² = D(x) with D(x) a square-free polynomial with coefficients in k. If D has even degree, or if D has odd degree and the leading coefficient of D is a non-square in k, then the Jacobian of the hyperelliptic curve y² = D(x) is essentially isomorphic to the ideal class group of the ring k[x,y]. This is the finite Abelian group of fractional ideals of k[x,y] modulo principal fractional ideals. Although generically, the 3-Sylow subgroup of this ideal class group is small (and frequently trivial), it is possible to generate hyperelliptic function fields – even infinite families of such fields – whose 3-rank is unusually large. This talk presents several methods for explictly constructing hyperelliptic function fields of high 3-rank, and more generally, high l-rank for any prime l coprime to the characteristic of k. Some of these techniques are adapted from constructions originally proposed for quadratic number fields by Shanks, Craig, and Diaz y Diaz, while others are specific to the function field setting. In particular, we explore how extending the field of constants k can lead to an increase in the 3-rank of the hyperelliptic function field.

This is joint work with Mark Bauer and Mike Jacobson, both of the University of Calgary, and Yoonjin Lee of Ewha Women's University in Seoul, South Korea.