Construction of hyperelliptic function fields of high three-rank

We present several explicit constructions of hyperelliptic function fields whose Jacobian or ideal class group has large 3-rank. Our focus is on finding examples for which the genus and the base field are as small as possible. Most of our methods are adapted from analogous techniques used for generating quadratic number fields whose ideal class groups have high 3-rank, but one method, applicable to finding large l-ranks for odd primes l ≥ 3, is new and unique to function fields. Algorithms, examples, and numerical data are included.