Computational Arithmetic Geometry

Sydney, June 18 - 20, 2003

Abstracts of the talks

Author: Nils Bruin (University of Sydney)
Title: Prym varieties of smooth plane quartics
Abstract: In this talk I will investigate non-hyperelliptic curves of genus 3 that admit an unramified double cover. The covering curve can be embedded into an Abelian surface, the Prym variety of the cover. This embedding, can be made completely explicit and can be used to apply Chabauty-like techniques and to compute in part of the Brauer-Manin obstruction of the cover.

I will give an example of how one can use this construction to determine the rational points on a curve of genus 3 without having to compute the Mordell-Weil group of its Jacobian. I will also give an example of a curve of genus 3 with a double cover of genus 5 which both violate the Hasse principle.

Author: Michael Bush (University of Illinois)
Title: Class towers of number fields
Abstract: Since Golod and Shafarevich's work in the 1960's it has been known that there exist number fields with infinite $p$-class towers. In general though it is a difficult problem to determine whether a given field has a finite or infinite tower. In this talk we describe a method for computing large quotients of the Galois group associated to the tower. We will describe some of our results for $2$-class towers of imaginary quadratic fields. In some cases we can show the tower must be finite where this was unknown previously.

Author: Claus Fieker (University of Sydney)
Title: Constructions of Class Fields of Global Fields
Abstract: Since the advent of class field theory one of the main problems has been its abstract nature that prevented researchers from computing examples. Thus although class field theory classifies abelian extensions of global fields, it failed to provide explicit defining equations for them. Recent progress in computational number theory now permits us to close the gap.

In this talk I will explain how one can find explicit defining equations utilizing Kummer, Artin-Schreier and Witt theory. Applications of these techniques include the explicit construction of good linear codes.

Author: Martine Girard (University of Sydney)
Title: Computing sets of rational points using Dem'janenko-Manin's method
Abstract: Although it is a well-known fact that the set of rational points of a curve of genus greater than 1 is finite, there exist effective methods only in particular cases. One of these method --Dem'janenko-Manin -- applies when the curve admits m independent morphisms to an elliptic curve of rank less than m. We construct families of genus 3 curves satisfying this condition and determine their sets of rational points. This is joint work with Leopoldo Kulesz.

Author: Joost van Hamel (University of Sydney)
Title: The Brauer-Manin obstruction and S-units
Abstract: The Brauer-Manin obstruction is an obstruction to the local-global principle for rational points of varieties over number fields. For smooth projective curves it was observed by Victor Scharaschkin that this obstruction detects exactly the intersection of the adelic points of the curve with the profinite completion of the Mordell-Weil group inside the Jacobian. It is an open question whether this intersection coincides with the rational points of the curve, or whether at least this intersection is empty whenever the curve has no rational points.

In this talk I will discuss computer experiments in progress concerning the analogous, but computationally simpler problem of the local-global principle for (S-)integral points on affine curves of the form f(x)y = 1. Here the (generalised) Jacobian is an algebraic torus, and the Brauer-Manin obstruction detects the intersection of the adelic points of the curve with the profinite completion of the group of (S-)integral points of the torus. The computer experiments should provide an indication how close this intersection gets to the set of (S-)integral points of the affine curve.

Author: Bill Hart (Macquarie University, Sydney)
Title: Modular Equations for Weber-type Functions of Higher Signature
Abstract: The usual Weber functions $f, f_1$ and $f_2$ of signature two are known to satisfy modular equations of various degrees. Bill has managed to generalize these functions to signatures three, five and seven. He has also found that these new sets of functions satisfy modular equations. Bill will describe these modular equations and provide a simple algorithm for computing them without numerical approximation! He will explicitly give modular equations of low degree in these signatures and indicate why the method fails for higher signatures. Bill will also briefly speak about a remarkable link that exists between the theory for signature three and Ramanujan's alternative cubic theory of the hypergeometric function.

A preprint will be available at

Author: David Kohel (University of Sydney)
Title: Galois Module Structure and Ranks for Weierstrass Subgroups
Abstract: The subgroup of the Jacobian of an algebraic curve generated by its Weierstrass points is a geometric invariant of the curve. The Galois module structure of this finitely generated group is an arithmetic invariant which can be used to study the structure of this subgroup. In this talk I will report on joint work with Martine Girard, using irreducible Galois submodules to prove the maximality of the rank of the Weierstrass subgroup for the generic curves in known families of genus three.

Author: Victor Scharaschkin
Title: Elliptic Curves and the Weak ABC Conjecture
Abstract: An abc triple is a triple of positive integers (a,b,c) with a + b = c. Its quality is q = log c/ log rad(abc), where rad(n) is the product of distinct primes dividing n. The abc conjecture is the statement that as c approaches infty, lim sup q = 1. The weak abc conjecture is that 1 is a limit point.

Elkies' proposed using maps from curves to the projective line unramified outside 0,1, infty called Belyi maps to generate abc triples. We examine triples produced from elliptic curves in this way, and give cohomological conditions on the elliptic curves which would imply that the weak abc conjecture holds.

Author: William Stein (Harvard)
Title: Explicitly computing endomorphism rings of modular abelian varieties
Abstract: First I will describe a way to represent, on a computer, modular abelian varieties over number fields, morphisms between them, and elements of them. To focus on one problem, I will then discuss computation of Hom(A,B) as an explicitly given sub-module of the homomorphisms Hom(H_1(A,Z),H_1(B,Z)) of integral homology.

I'll present some material that was known long ago to Shimura, Ribet and others, and I'll point out some open problems. For example, efficiently and explicitly computing Hom(A,B) seems to be more difficult than I first expected, since the formulas in papers of Shimura and Ribet assume that one is working with Gamma1(N) with N large.

Author: Mark Watkins (Penn State)
Title: Elliptic Curves with lots of integral points
Abstract: We report on an implementation of an idea of Elkies to find elliptic curves with lots of integral points via finding *pairs* of integral points. While the naive algorithm takes H^9 time to find all curves within a search range of size H, the Elkies algorithm takes time H^8, but with a possibly bigger constant. We will explain how to make his algorithm workable in practise; the naive algorithm becomes cumbersome at H=26 or so, while we have run the pair-finding algorithm for H=40, and are starting on H=60. We will also give record-breaking curves for the lowest known conductor and absolute discriminant for ranks 9 and 10.

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