* Construction of Hyperelliptic Function Fields of High
Three-Rank*Renate Scheidler, University of Calgary

**Abstract:**

A hyperelliptic function field is a field of the form k(x,y) where k is a finite
field of odd characteristic and y^{2 }= 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 y^{2 }= 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 teachniques
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 Simon Fraser University.

PDF of slides