Automorphisms of minimal entropy on supersingular K3 surfaces

Abstract

In this article we give a strategy to decide whether the logarithm of a given Salem number is realized as entropy of an automorphism of a supersingular K3 surface in positive characteristic. As test case it is proved that $\log {\lambda }_d$, where ${\lambda }_d$ is the minimal Salem number of degree $d$, is realized in characteristic $5$ if and only if $d\le 22$ is even and $d\ne 18$. In the complex projective setting we settle the case of entropy $\log {\lambda }_{12}$ left open by McMullen, by giving the construction. A necessary and sufficient test is developed to decide whether a given isometry of a hyperbolic lattice, with spectral radius bigger than one, is positive, i.e. preserves a chamber of the positive cone.

arXiv comment26 pages, added matrices representing the automorphisms as ancillary data

References in corpus (5)

• The Tate conjecture for K3 surfaces over finite fields
• Supersingular K3 surfaces for large primes
• 2-adic integral canonical models and the Tate conjecture in characteristic 2
• Elliptic fibrations on K3 surfaces and Salem numbers of maximal degree
• Dynamics on supersingular K3 surfaces and automorphisms of Salem degree 22

Cited by in corpus (3)

• Minimum positive entropy of complex Enriques surface automorphisms
• On the stable dynamical spectrum of complex surfaces
• The non-symplectic index of supersingular K3 surfaces