Computational Algebra Seminar: Scheidler -- Construction of Hyperelliptic Function fields of High Three-Rank

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 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.