Banff, 4 - 9 February, 2007
Organizers: Bjorn Poonen, Nils Bruin
At the request of the organizers, I will introduce/review some constructions from p-adic Hodge theory that intervene in the usual Chabauty method; these include the comparison isomorphism between the de Rham and etale cohomology groups of a curve, and the Bloch-Kato exponential map. I will focus on the case of good ordinary reduction, where these constructions can be made reasonably explicit. The goal is to analogize the explicit descriptions to the higher unipotent de Rham and etale fundamental groups, in a manner useful for doing nonabelian Chabauty; as time and my abilities permit, I will start doing this (again only in the good reduction case) using some work of Martin Olsson.
We use a Chabauty computation to determine the set of rational points on a curve of higher genus. The input for a Chabauty computation includes the Mordell-Weil rank of the associated Jacobian. Traditionally we bound, and hope to determine, the Mordell-Weil rank using a Selmer group. In this talk, we will survey the methods for computing a Selmer group of a Jacobian using functions on the curve. We will review both major methods. The first is quite general, but is inefficient for cyclic covers of the projective line (like hyperelliptic curves). The second method addresses such covers.
We will discuss how one can obtain information on rational points by combining coverings with local information. We will focus on the case of abelian coverings and explain the relationship with the Brauer-Manin obstruction. If explicit generators of the Mordell-Weil group are known, this can be implemented efficiently, leading to a procedure known as the Mordell-Weil sieve. We will formulate a conjecture that, if valid for a given curve, implies that we can effectively decide whether a given coset of N times the Mordell-Weil group meets the image of the curve or not. If we know that each such coset contains at most one point coming from the curve, this means that we can determine the set of rational points on the curve.
Let C be a curve of genus g >= 3 and let C^{(d)} denote its d-th symmetric power. We explain an adaptation of Chabauty which allows us in many cases to compute C^{(d)}(Q) provided the rank of the Mordell-Weil group is at most g-d. Cases for which our method should work include:
Our adaptation of Chabauty differs from the classical Chabauty in that we combine Chabauty type information given by several primes.
Example. Let C be the genus 3 hyperelliptic curve \begin{equation}\label{eqn:C} C: y^2=x(x^2+2)(x^2+43)(x^2+8x-6) \end{equation} with Jacobian having rank $1$. Let $\pi:C\rightarrow P^1$ be the $x$-coordinate map. We show that $C^{(2)}(\Q)$ consists of $\pi^{-1} P^1(\Q)$ plus $10$ other points which we write down explicitly. Here we needed to combine the Chabauty information at primes $p=5$, $7$, $13$. It is noteworthy that $C^{(2)}$ in this example is a surface of general type.
It is well-known that not all varieties over Q satisfy the Hasse principle. The famous Selmer curve given by 3x^3+4y^3+5z^3=0 in P^2, for instance, indeed has points over every completion of Q, but no points over Q itself. Though it is trivial to find points over some cubic field, it is a priori not obvious whether there are points over a cubic field that is galois. We will see that such points do exist. K3 surfaces do not satisfy the Hasse principle either, which in some cases can be explained by the so called Brauer-Manin obstruction. It is not known whether this obstruction is the only obstruction to the existence of rational points on K3 surfaces. We relate the two problems by sketching a proof of the following fact. If there exists a smooth curve over Q given by ax^3+by^3+cz^3=0 that is locally solvable everywhere, that has no points over any cubic galois extension of Q, and whose Jacobian has trivial Mordell-Weil group, then the algebraic part of the Brauer-Manin obstruction is not the only one for K3 surfaces. No knowledge about K3 surfaces or Brauer-Manin obstructions will be assumed as known.
We explain a correspondence between triliear forms and triples of genus one curves with a fixed Jacobian and some added structure. We generalize the addition law on elliptic curves to addition on certain "cubes" of numbers. We explain how this works for arbitrary rings, and we give a natural construction of points on elliptic curves to other points on other elliptic curves which generalizes a known construction from class field theory.
I will discuss theoretical and computational results toward the following problem: given a specific elliptic curve over Q, compute the exact order and structure of its Shafarevich-Tate group in practice. I view this problem as a motivating question for organizing both theoretical and algorithmic investigations into the arithmetic of elliptic curves and the Birch and Swinnerton-Dyer conjecture.
The height of a rational point on an elliptic curve measures the size of the point. The enormous gap between the lower and upper bound (Lang's conjectures) of the height of such a point, prompted the comparison of the elliptic curve scenario with that of the multiplicative group, the Brauer-Siegel theorem. In this talk, a conjectural Brauer-Siegel theorem for elliptic curves over the rationals will be discussed and interesting questions which arise in this context motivated by computation will be presented.
We considered all genus 2 curves y^2 = f(x) where f has integral coefficients of absolute value at most 3; there are about 200,000 isomorphism classes of such curves. Using various methods (point search, local solubility, 2-descent, Mordell-Weil sieve), we attempted to decide for each curve whether it possesses rational points. In all but 42 cases, we were successful; in the remaining cases, our result is conditional on the Birch and Swinnerton-Dyer conjecture. In the talk, we will explain the methods we used and the improvements we came up with, and dicuss the results.
