Here is a list of my papers followed by postscript files of eight lectures.

2-descent on the Jacobians of hyperelliptic curves, Journal of Number Theory, (51), 1995, 219-232.

Class groups and Selmer groups, Journal of Number Theory, (56), 1996, 79-114. This paper gives bounds on the index of the intersection of a Selmer group and a quotient of the dual of part of a class group in each of the two groups.

Arithmetic and geometry of the curve 1+y^3 = x^4 (with M.J. Klassen), Acta Arithmetica, (74), 1996, 241-257. This paper shows that the set of Weierstrass points (the flexes) is the same as the set of rational points over the field Q(zeta_12) and is a torsion packet. It also finds the bitangents to this curve and bases for the 2- and 3-torsion of the Jacobian.

A simplified Data Encryption Standard algorithm, Cryptologia, (20), 1996, 77-84. This paper gives a method of explaining the DES algorithm to a cryptography class.

Cycles of quadratic polynomials and rational points on a genus-two curve, (with E.V. Flynn & B. Poonen), Duke Mathematical Journal, (90), 1997, 435-463. This paper shows that there are no rational numbers that have period length five with respect to a rational quadratic polynomial in one variable. To do this required finding the set of rational points on a genus 2 curve. This involved a 2-descent and a Chabauty-style argument.

Computing a Selmer group of a Jacobian using functions on the curve, Mathematische Annalen, (310), 1998, 447-471. This paper gives a general algorithm for finding Selmer groups for the Jacobians of curves. It includes discussions of the assumptions such algorithms seem to be based on. The Selmer group is for an isogeny, over a number field, from an abelian variety to the Jacobian of a curve where the kernel of the isogeny is killed by a power of a prime.

Explicit descent for Jacobians of cyclic covers of the projective line (with B. Poonen), Journal fuer die Reine und Angewandte Mathematik, (488), 1997, 141-188. This paper gives an algorithm for finding the 1-zeta_p Selmer group for the Jacobian of a curve y^p = f(x).

Computing the p-Selmer group of an elliptic curve (with Z. Djabri and N. Smart). To appear in Transactions of the American Mathematical Society. This paper gives a practical algorithm for finding a group containing the p-Selmer group for an elliptic curve where p is an odd prime. In all worked examples, this group actually is the p-Selmer group.

Emprical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves (with F. Lepre'vost, V. Flynn, W. Stein, M. Stoll and J. Wetherell). To appear in Mathematics of Computation.

Talk at pre-ANTS, June 28, 2000

Click on Can the 5-part of the Shafarevich-Tate group of an elliptic curve be arbitrarily large? to download a postscript file of the lecture.

Intercity Number Theory Seminar lecture series

Here are the postscript files of the 7 talks I gave in the Intercity Number Theory Seminar in early 1999. There are abstracts below. I am grateful to the Intercity Number Theory Seminar and the Mathematisch Instituut at the Rijksuniversiteit Leiden for their hospitality during my visit.

Click on Lecture 1 notes to download a postscript file of the first lecture's notes.

Click on Lecture 2 notes to download a postscript file of the second lecture's notes.

Click on Lecture 3 notes to download a postscript file of the third lecture's notes.

Click on Lecture 4 notes to download a postscript file of the fourth lecture's notes.

Click on Lecture 5 notes to download a postscript file of the fifth lecture's notes.

Click on Tamagawa numbers to download a postscript file of the first talk at the workshop "Advances in number theory".

Click on Neron models to download a postscript file of the second talk at the workshop "Advances in number theory".

Summary of the first five lectures.

Let us say that we are interested in finding all of the points with rational coordinates on an algebraic curve. One method that has seen much development in the past decade is the following. To the algebraic curve we can associate a Jacobian. This is a group variety whose dimension is the same as the genus of the curve and can also be considered to be the divisor classes of degree 0 of the curve. Roughly speaking, the Jacobian of an elliptic curve is itself. The rational part of the Jacobian is a finitely generated abelian group called the Mordell-Weil group. It is usually straightforward to find the torsion subgroup of the Mordell-Weil group, but we do not yet have an effective algorithm for finding the free Z-rank of the group, known as the Mordell-Weil rank. The structure of the Mordell-Weil group needs to be known before generators for this group can be found. With generators, there are sometimes techiques for determining the set of points with rational coordinates on the underlying curve. In this series of lectures, we will discuss techniques that often enable us to find the Mordell-Weil rank. We will motivate this discussion with problems like finding the longest string of squares in arithmetic progression (a string of three is 1, 25, 49). Along the way, we will discuss the techniques from Galois cohomology and algebraic geometry, that we will need.

Abstract of the first lecture.

In this first talk, we will define Mordell-Weil groups. We will motivate interest in them by solving the following problem (first solved by Mordell in the 1960's): when is the product of two consecutive integers equal to the product of three consecutive integers? We will be able to solve the problem, modulo computation of the Mordell-Weil group. Hopefully this will convince the audience of the desireability of being able to compute such groups. In a later lecture, we will compute the structure of the Mordell-Weil group for this case, and thus complete the solution of this problem.

Abstract of the second lecture.

In this second talk, we will first define Jacobians. In attempting to determine the structure of the Mordell-Weil group of a Jacobian, we are drawn into Galois cohomology. We will have a short course on Galois cohomolgy, covering just enough to get by. We will discuss Jacobians over local fields. Putting the local cohomological information together, we get a Selmer group. We will define them and show why they are finite. Selmer groups give an upper bound on the free Z-rank of a Mordell-Weil group.

Abstract of the third lecture.

In this third talk, we will show that the Selmer group is finite. Then we will work to replace the cohomolgy groups in the diagram used to define a Selmer group with groups in which it is easier to work.

Abstract of the fourth lecture.

In this fourth talk, we will start by showing that the map F, coming from functions on the curve is indeed relating to a coboundary map, a map induced by a Weil-pairing and a Kummer map. We will apply the algorithm developed thus far to cyclic covers of the projective line with a rational Weierstrass point. This class of curves includes elliptic curves.

Abstract of the fifth lecture. In this fifth talk, we resolve the three motivating questions from the first talk. The first problem was "When is the product of two consecutive integers equal to the product of three consecutive integers?" We will show that the Mordell-Weil rank of the associated elliptic curve is one. The next question was to find the longest sequence of squares in arithmetic progress. One such sequence is 1, 25, 49. We will show that sequences of four squares in arithmetic progression lead to points on an elliptic curve. We will find the Mordell-Weil group of that elliptic curve and resolve the question. (I don't want to spoil the surprise). Lastly we will say what we can about solutions of y^2=x^5+1 in number fields.

Abstract of the Tamagawa numbers lecture.

In this talk, we will discuss algorithms for finding the minimal, proper, regular model for a curve over a p-adic field and its special fiber (the funny stick figure). We then use that to find the structure of the component group of the Neron model of its Jacobian: J(Qp)/J_0(Qp). We give an example starting with a genus 2 curve.

Abstract of Neron models lecture

In this talk, we define Neron models and give some applications.

Click on homepage to get back to my homepage.

I recommend the University of Georgia at Athens' number theory site.

Last changed: 2-Jun-1999