It has been conjectured that for N sufficiently large, there are no quadratic polynomials in Q[z] with rational periodic points of period N. Morton proved there were none with N=4, by showing that the genus 2 algebraic curve that classifies periodic points of period 4 is birational to X$_1$(16), whose rational points had been previously computed. We prove there are none with N=5. Here the relevant curve has genus 14, but it has a genus 2 quotient, whose rational points we compute by performing a 2-descent on its Jacobian and applying a refinement of the method of Chabauty and Coleman. We hope that our computation will serve as a model for others who need to compute rational points on hyperelliptic curves. We also describe the three possible Galois-stable 5-cycles, and show that there exist Galois-stable N-cycles for infinitely many N. Furthermore, we answer a question of Morton by showing that the genus 14 curve and its quotient are not modular. Finally, we mention some partial results for N=6.

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Cycles of quadratic polynomials and rational points on a genus-$2$ curve

The classification of rational preperiodic points of quadratic polynomials over ${\Bbb Q}$ : a refined conjecture

Canonical Heights on Projective Space

The real and complex dynamics of the H6non map 4) : Ik2--*~ 2, 4 ) ( x , y ) = ( y , y 2 + b + a x ) (1) have been extensively studied since H6non [6] showed that such maps may have strange attractors. See for example [1, 3, 4, 6, 8, 12]. In this paper we are going to investigate some of the algebro-geometric and number theoretic properties of the H6non map. The major difficulty in using algebraic geometry to study the H6non map is that the map does not extend in a natural way to a morphism of the projective plane IP 2. Thus if we homogenize the equation for 4) we find that the resulting map 4): m E ~ IP 2, 4)([x, Y , Z ] ) = [ r z , y2 + bZ 2 + a X Z , Z 23 , (2) is not defined at the point [1,0,0] . It is a rational (in fact, birational) map on IP 2, but it is not a morphism. Similarly the inverse map

D.J Lewis

Papers

Invariant sets of morphisms on projective and affine number spaces