July 4-10, 2010
Rational Points 3
Workshop at Schloss Thurnau
Elliptic curves E and F are n-congruent if their n-torsion subgroups are isomorphic as Galois modules via an isomorphism respecting the Weil pairing. Rubin and Silverberg gave formulae for the families of elliptic curves n-congruent to a given elliptic curve for n = 2,3,4,5. Formulae in the case n = 7 are given by Halberstadt and Kraus. I will describe an invariant-theoretic approach to obtaining these formulae and use it to extend to the cases n = 9 and n = 11. There are corresponding formulae when the isomorphism of n-torsion subgroups does not respect the Weil pairing.
Let C be a curve of genus at least 2 over a number
field K of degree d. Let J
be the Jacobian of C and r the rank of the Mordell-Weil
group J(K). Chabauty is a practical method for explicitly
computing C(K) provided
r ≤ g-1.
In unpublished work, Wetherell suggested that Chabauty’s method should still be applicable provided the weaker bound r ≤ d(g-1) is satisfied. We give details of this and use it to solve the Diophantine equation x2 + y3 = z10 by reducing the problem to determining the K-rational points on several genus 2 curves over K = Q(21/3).
We present computational results on computing the divisor class number of a cubic function field over a large base field. The underlying method is based on well-known approximation methods of the Euler product representation of the zeta function of such a field. Those methods proved to be very efficient in number fields and were used to verify several heuristics Our implementation provides numerical evidence of the computational effectiveness of this algorithm for cubic function fields. The examples provided are the largest divisor class numbers and regulators ever computed for a (random) cubic function field over a large prime field. The ideas underlying the optimization of the class number algorithm can in turn be used to analyze the distribution of the zeros of the function field's zeta function. We provide a variety of data on a certain distribution of the divisor class number that verify heuristics by Katz and Sarnak on the distribution of the zeroes of the zeta function.
The talk will present results from joint work with Florian Pop, in which we found that sections of the fundamental group extension of a hyperbolic curve over a p-adic field all meet certain geometric requirements.
I will present a theory of explicit descent that includes and generalizes most of the versions of explicit descent that have been used in practice, with an eye towards applications to be discussed in the following lecture by Nils Bruin.
I shall give a progress report on an ongoing joint project with Bjorn Poonen and Michael Stoll to compute reasonable approximations of 2-Selmer groups of Jacobians of smooth plane quartics. As a test case, we tried to apply our methods to a list of plane quartics with small discriminant, provided by Denis Simon. In the process we found some surprising torsion structures and some interesting other features. I shall discuss our findings.
We discuss the question of whether the Brauer-Manin obstruction to the Hasse principle is the only one for integral points on affine hyperbolic curves. We give several equivalent formulations of this question and relate it to an old conjecture of Skolem. This is a joint work with J.F. Voloch.
For a smooth and projective variety over a number field with torsion free geometric Picard group and finite transcendental Brauer group we show that only the archimedean places, the primes of bad reduction and the primes dividing the order of the transcendental Brauer group can turn up in the description of the Brauer-Manin set. (This is a joint work with Colliot-Thélène.)
We give a construction of an Enriques surface over Q with an étale-Brauer obstruction to the Hasse principle and with non-empty algebraic Brauer set. We also give evidence that this obstruction is not explained by a transcendental Brauer-Manin obstruction. This is joint work with Anthony Várilly-Alvarado.
It is well-known that K3 surfaces over number fields need not satisfy the Hasse principle or weak approximation. All known counter-examples to date, however, involve K3 surfaces that are endowed with an elliptic fibration structure; in fact, the fibration is essential to the computation of Brauer classes that reveal obstructions to the Hasse principle and weak approximation. General K3 surfaces, i.e., K3 surfaces with geometric Picard rank one, do not enjoy this kind of structure. I will explain how to construct certain K3 surfaces of geometric Picard rank one, together with a transcendental quaternion algebra that obstructs weak approximation of rational points. This is joint work with Brendan Hassett and Patrick Várilly-Alvarado.
Michael Stoll (Universität Bayreuth)