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Computational Arithmetic Geometry

Vancouver, July 5 - 9, 2004

Abstracts of the PIMS Distinguished Visitor lectures

Author: Bjorn Poonen (UC Berkeley)
Title: Rational points on varieties
Abstract: I will discuss what is known and conjectured about computing the set of rational solutions to a system of polynomial equations.

Author: Bjorn Poonen (UC Berkeley)
Title: Torsors and descent
Abstract: Around 1930, Chevalley and Weil realized that the reduction step in Fermat's method of infinite descent could be applied whenever one had a finite unramified morphism of varieties X --> Y. Their idea has proved to be indispensable for the determination of the set of rational points on curves. In the case where X is geometrically Galois over Y with Galois group G, one can view X also as a G-torsor over Y.

In the 1980s, Colliot-Thélène and Sansuc showed how the Chevalley-Weil approach could be generalized to G-torsors for certain algebraic groups G even when G is not zero-dimensional. I will give an introduction to their theory.

Author: Bjorn Poonen (UC Berkeley)
Title: Diagonal cubic surfaces and the magic square torus
Abstract: Above certain surfaces, there are no nontrivial unramified covers but there are G-torsors for some G of positive dimension. In these cases, the Chevalley-Weil descent is useless, but the theory of Colliot-Thélène and Sansuc sometimes succeeds in determining the rational points. I will discuss a conjectural application of this theory to the case of projective surfaces defined by homogeneous equations of the form
a x^3 + b y^3 + c z^3 + d w^3 = 0
where a,b,c,d are nonzero integers.

Abstracts of the talks

Author: Michael Bennett (University of British Columbia)
Title: Diophantine equations via modular methods
Abstract: TBA

Author: Martin Bright (University of Liverpool)
Title: Computing the Brauer-Manin Obstruction
Abstract: TBA

Author: Nils Bruin (Simon Fraser University)
Title: The arithmetic of Prym varieties in Genus 3
Abstract: The theory of Prym-varieties for hyperelliptic curves can be approached via Kummer theory. In combination with Chabauty-techniques it has given very practical methods to bound the number of rational points on hyperelliptic curves.

The simplest non-hyperelliptic curves are of genus 3. We will discuss how the theory of Prym-varieties can be made effective for non-hyperelliptic curves of genus 3 and how this can be applied to a variety of problems. As an example, we will determine the rational points on a curve of genus 3 with no specific geometric properties, without computing the Mordell-Weil group of its Jacobian. We will also give an example of a curve of genus 3 and a curve of genus 5 that violate the Hasse-principle and we will show how one can compute part of the Brauer-Manin obstruction of a genus 5 curve embedded in an Abelian surface.

Author: Iftikhar Burhanuddin
Title: Computing rational torsion on elliptic curves in linear time
Abstract: We present an algorithm to decide whether a elliptic curve over $Q_p$ has a non-trivial $p$-torsion part ($\#E(Q_p)[p] \neq 1$) under certain assumptions. We use this algorithm to efficiently determine $E(Q)_{tors}$ the group of $Q$-rational torsion points on an elliptic curve.

This is joint work with Prof. Ming-Deh Huang.

Author: Victor Flynn (University of Liverpool)
Title: Shafarevich-Tate groups of Jacobians
Abstract: I shall describe joint work with Nils Bruin, giving an approach to computing in the Shafarevich-Tate group of Jacobians of higher genus curves, with an emphasis on the theory and practice of visualisation via number fields. This allows widely applicable techniques for the computation of ranks of Jacobians of higher genus curves, even when the 2-Selmer bound is not attained; this was previously only possible for a few special cases.

I shall also describe a connection with degree 4 del Pezzo surfaces, and show how the Brauer-Manin obstruction on these surfaces can be used to compute members of the Shafarevich-Tate group of Jacobians. I shall derive an explicit parametrised infinite family of genus 2 curves whose Jacobians have nontrivial members of the Shafarevich-Tate group.

Author: Ed Schaefer (Santa Clara University)
Title: p-Selmer groups that can be arbitrarily large
Abstract: In this talk, we will present a proof of the following result: Let p be a prime, at least 5. Let m be a positive integer. Then there is an elliptic curve E over a number field K where the p-Selmer group has size at least p^m and where the degree of K over Q is at most genus(X_0(p)) + 1. This is joint work with Remke Kloosterman, Rijksuniversiteit Groningen.

Author: Tony Shaska (University of Idaho)
Title: On the field of moduli of curves
Abstract: Let $k$ be an algebraically closed field of characteristic zero and $\X_g$ a genus $g$ projective irreducible algebraic curve defined over $k$. The {\it field of moduli} is the field of definition of the representing point $\p=[\X_g]$ in the moduli space $\M_g$. For an algebraic curve the field of moduli is not in general a field of definition. Investigating the obstruction is part of descent theory for fields of definition and has many consequences in arithmetic geometry. Many works have been devoted to this problem, most notably by Weil, Shimura, and Grothendieck, among many others. The main goal of this talk is to investigate for which curves the field of moduli is a field of definition. The Weil criterion assures that if a curve has no automorphisms then its field of moduli is a field of definition. Hence, we focus on curves which have nontrivial automorphism group. Our main conjecture says that in the variety of moduli of hyperelliptic curves the obstruction occurs only for curves with automorphism group $\Z_2$. We will briefly describe some known results in the literature and some recent developments.

Author: Ben Smith (University of Sydney)
Title: Realising endomorphisms of Jacobians with correspondences
Abstract: We apply the theory of correspondences on curves to give representations and algorithms for concrete computations with isogenies of jacobians varieties of curves, with a particular emphasis on computing in the endomorphism ring of the jacobian of an algebraic curve.

Author: Alexa van der Waall (Simon Fraser University)
Title: Differential rings and operators in Magma
Abstract: In the newest release of Magma, machinery for defining and working with differential rings and fields has been implemented. This includes constructing field extensions, quotient rings and fields of fractions.

In addition, structures are constructed to create differential operators over a differential fields. Calculations related to differential operators, such as determining the singular points and the determination of rational solutions of a linear differential operator are implemented.

In this talk we discuss some of the new features of differential rings and operators, as in Magma 2.11, and demonstrate them on the spot via some examples.

Author: Soroosh Yazdani (UC Berkeley)
Title: Diophantine equations of signature (n,n,3).
Abstract: In this talk, I will describe how one can use the techniques used to prove Fermat's Last Theorem, to prove similar Diophantine equations such as $x^n+y^n=z^3$.

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