Automorphisms of Elliptic K3 surfaces and Salem numbers of maximal degree

Abstract

Using elliptic structures, we show that any supersingular K3 surface of Artin invariant 1 in characteristic p = 5, 7, 13 has an automorphism the entropy of which is the natural logarithm of a Salem number of degree 22.

Equidistribution problems in complex dynamics of higher dimension

“…Note that when f is algebraically expanding, then its action on cohomology is simple because dim H k,k (X, C) = 1. Finally, though analytic tools are dominant in the study of the objects introduced in this section, many questions are of algebraic nature and can be asked for maps or correspondences which are defined over fields different from C. The reader will find in [40,41,76] and the references therein some recent algebraic counterparts of the topics presented in this paper.…”

Automorphisms of minimal entropy on supersingular K3 surfaces

“…It turns out we can glue this to (E 6 , h) where h ∈ O(E 6 ) is given by the central symmetry of the Dynkin diagram of E 6 like for λ 16 (see Figure 6). Now (S ⊕ φ E 6 , f 0 ⊕ h) is a lattice of signature (1,13) and discriminant group F 8 5 . Since 5 is prime in O K , s 8 is the irreducible characteristic polynomial of the action on the discriminant group.…”

Dynamics on supersingular K3 surfaces

“…We point out that the automorphisms in Theorem 3.5 are mostly implicit, i.e. there are existence results building on elliptic fibrations (in particular of maximal rank) and some group theory (see [2,4]). In contrast, until the completion of the first version of this paper it was only for p D 3 that there was an explicit g 2 Aut.X.p// of Salem degree 22 known [3], building on the calculation of Aut.X.3// by Kondō and Shimada [8].…”

“…One of the keys for the implicit results in [2,4] is the special feature that a K3 surface may admit different elliptic fibrations. For convenience, we shall only work with 714 M. Schütt CMH fibrations which are already visible on X .…”

“…Esnault and Oguiso illustrated this phenomenon explicitly with the Fermat quartic in characteristic 3, building on work of Kondō and Shimada [8]. On the implicit side, there are nonliftability results for the supersingular K3 surface X.p/ of Artin invariant D 1 in characteristic p ¤ 5; 7; 13 in [2,4]. Our aim is to exhibit non-liftable automorphisms on an infinite series of supersingular K3 surfaces in an explicit and systematic manner: The key ingredient of our approach consists in the theory of elliptic fibrations on K3 surfaces and in particular Mordell-Weil lattices, since these are at the same time accessible, versatile and allow for explicit descriptions of automorphisms.…”