Nils Bruin

Two-cover descent on hyperelliptic curves

We describe an algorithm that determines a set of unramified covers of a given hyperelliptic curve, with the property that any rational point will lift to one of the covers. In particular, if the algorithm returns an empty set, then the hyperelliptic curve has no rational points. This provides a relatively efficiently tested criterion for solvability of hyperelliptic curves. We also discuss applications of this algorithm to curves of genus 1 and to curves with rational points.

Deciding Existence of Rational Points on Curves: An Experiment

In this paper we gather experimental evidence related to the question of deciding whether a curve has a rational point. We consider all genus-2 curves over ℚ given by an equation y 2 = f(x) with f a square-free polynomial of degree 5 or 6, with integral coefficients of absolute value at most 3. For each of these roughly 200 000 isomorphism classes of curves, we decide whether there is a rational point on the curve by a combination of techniques that are applicable to hyperelliptic curves in general. In order to carry out our project, we have improved and optimized some of these techniques. For 42 of the curves, our result is conditional on the Birch and Swinnerton-Dyer conjecture or on the generalized Riemann hypothesis.

Powers from Products of Consecutive Terms in Arithmetic Progression

We show that if

Towers of 2-covers of hyperelliptic curves

k

On Powers as Sums of Two Cubes

is a positive integer, then there are, under certain technical hypotheses, only finitely many coprime positive

GENERALIZED EXPLICIT DESCENT AND ITS APPLICATION TO CURVES OF GENUS 3

k

The arithmetic of Prym varieties in genus 3

-term arithmetic progressions whose product is a perfect power. If

The Brauer-Manin Obstruction and Sha[2].

4 \leq k \leq 11

Some ternary Diophantine equations of signature (n, n, 2)

, we obtain the more precise conclusion that there are, in fact, no such progressions. Our proofs exploit the modularity of Galois representations corresponding to certain Frey curves, together with a variety of results, classical and modern, on solvability of ternary Diophantine equations. As a straightforward corollary of our work, we sharpen and generalize a theorem of Sander on rational points on superelliptic curves.

Visualising Sha[2] in Abelian surfaces

In this article, we give a way of constructing an unramified Galois-cover of a hyperelliptic curve. The geometric Galois-group is an elementary abelian 2-group. The construction does not make use of the embedding of the curve in its Jacobian, and it readily displays all subcovers. We show that the cover we construct is isomorphic to the pullback along the multiplication-by-2 map of an embedding of the curve in its Jacobian. We show that the constructed cover has an abundance of elliptic and hyperelliptic subcovers. This makes this cover especially suited for covering techniques employed for determining the rational points on curves. In particular the hyperelliptic subcovers give a chance for applying the method iteratively, thus creating towers of elementary abelian 2-covers of hyperelliptic curves. As an application, we determine the rational points on the genus 2 curve arising from the question of whether the sum of the first n fourth powers can ever be a square. For this curve, a simple covering step fails, but a second step succeeds.