Finding Rational Periodic Points on Wehler K3 Surfaces

New Zealand Journal of Mathematics

Vol. 39, (2009), Pages 133-141

Benjamin Hutz

Department of Mathematics and Computer Science

Amherst College

Amherst, MA

Abstract This article examines dynamical systems on a class of K3 surfaces in $\mathbb{P}^{2} \times \mathbb{P}^{2}$ with an infinite automorphism group. In particular, this article develops an algorithm to find $\mathbb{Q}$-rational periodic points using information modulo p for various primes p. The algorithm is applied to exhibit K3 surfaces with $\mathbb{Q}$-rational periodic points of primitive period $1,\ldots,16$. A portion of the algorithm is then used to determine the Riemann zeta function modulo 3 of a particular K3 surface and find a family of K3 surfaces with Picard number two.

Keywords periodic points, K3 surfaces, dynamical systems.

Classification (MSC2000) Primary: 11G99, 14G99. Secondary: 37F99.