39,173 research outputs found
Coverings of curves of genus 2
We shall discuss the idea of finding all rational points on a curve C by first finding an associated collection of curves whose rational points cover those of C. This classical technique has recently been given a new lease of life by being combined with descent techniques on Jacobians of curves, Chabauty techniques, and the increased power of software to perform algebraic number theory. We shall survey recent applications during the last 5 years which have used Chabauty techniques and covering collections of curves of genus 2 obtained from pullbacks along isogenies on their Jacobians
On symmetric square values of quadratic polynomials
We prove that there does not exist a non-square quadratic polynomial with
integer coefficients and an axis of symmetry which takes square values for N
consecutive integers for N=7 or N >= 9. At the opposite, if N <= 6 or N=8 there
are infinitely many
Covering collections and a challenge problem of Serre
We answer a challenge of Serre by showing that every rational point on the projective curve X + Y = 17 Z is of the form (1, 2, 1) or (2, 1, 1). Our approach builds on recent ideas from both Nils Bruin and the authors on the application of covering collections and Chabauty arguments to curves of high rank. This is the only value of c81 for which the Fermat quartic X + Y = c Z cannot be solved trivially, either by local considerations or maps to elliptic curves of rank 0, and it seems likely that our approach should give a method of attack for other nontrivial values of c
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