ACAC Seminar Abstract

ACAC Seminar Abstract

ACAC Seminars

ACAC Seminar Abstract

Construction of Hyperelliptic Function fields of High Three-Rank

Speaker: Renate Scheidler (joint work with M. Bauer, M. Jacobson and Y. Lee)
Date, Time: Fri, 18 Jan 2008 15:00

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.

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