Non-symplectic automorphisms of order 3 on K3 surfaces

In this paper we study K3 surfaces with a non-symplectic automorphism of order 3. In particular, we classify the topological structure of the fixed locus of such automorphisms and we show that it determines the action on cohomology. This allows us to describe the structure of the moduli space and to show that it has three irreducible components.     

The modularity of K3 surfaces with non-symplectic group actions

We consider complex K3 surfaces with a non-symplectic group acting trivially on the algebraic cycles. Vorontsov and Kondō classified those K3 surfaces with transcendental lattice of minimal rank. The purpose of this note is to study the Galois representations associated to these K3 surfaces. The rank of the transcendental lattices is even and varies from 2 to 20, excluding 8 and 14. We show that these K3 surfaces are dominated by Fermat surfaces, and hence they are all of CM type. We will establish the modularity of the Galois representations associated to them. Also we discuss mirror symmetry for these K3 surfaces in the sense of Dolgachev, and show that a mirror K3 surface exists with one exception.     

On Vorontsov's Theorem on K3 surfaces with non-symplectic group actions

We shall give a proof for Vorontsov's Theorem and apply this to classify log Enriques surfaces with large prime canonical index.

     

K3 surfaces with interesting groups of automorphisms

By the fundamental result of I. I. Piatetsky-Shapiro and I. R. Shafarevich (1971), the automorphism group Aut(X) of aK3 surfaceX over and its action on the Picard latticeS _X are prescribed by the Picard latticeS _X . We use this result and our method (1980) to show the finiteness of the set of Picard latticesS _X of rank 3 such that the automorphism group Aut(X) of theK3 surfaceX has a nontrivial invariant sublatticeS ₀ inS _X where the group Aut(X) acts as a finite group. For hyperbolic and parabolic latticesS ₀, this has been proved by the author before (1980, 1995). Thus we extend these results to negative sublatticesS ₀.We give several examples of Picard latticesS _X with parabolic and negativeS ₀.We also formulate the corresponding finiteness result for reflective hyperbolic lattices of hyperbolic type over purely real algebraic number fields. We give many examples of reflective hyperbolic lattices of the hyperbolic type.These results are important for the theory of Lorentzian Kac-Moody algebras and mirror symmetry.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 45, Algebraic Geometry-8, 1997.     

Classification of non‐symplectic automorphisms of order 3 on K3 surfaces

In this paper, we study non‐symplectic automorphisms of order 3 on algebraic K3 surfaces over ${\bb C}$ which act trivially on the Néron‐Severi lattice. In particular we shall characterize their fixed loci in terms of the invariants of 3‐elementary lattices. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Order 9 automorphisms of K3 surfaces

Abstract

In this paper, we provide a complete classification of non-symplectic automorphisms of order 9 of complex K3 surfaces.     

K3 surfaces with an order 50 automorphism

In any characteristic p different from 2 and 5, Kondō gave an example of a K3 surface with a purely non-symplectic automorphism of order 50. The surface was explicitly given as a double plane branched along a smooth sextic curve. In this note we show that, in any characteristic $p\ne 2,5$, a K3 surface with a cyclic action of order 50 is isomorphic to the example of Kondō.     

Regular maps of 2-power order

Journal of Algebraic Combinatorics - In this paper, we consider the possible types of regular maps of order $$2^n$$ , where the order of a regular map is the order of automorphism group of the map....     

Almost fixed-point-free automorphisms of order 2

Let φ be an automorphism of order 2 of the group G with C _G (φ) finite. We prove the following. If G has finite Hirsch number then G is (nilpotent of class at most 2)-by-finite but need not be abelian-by-finite. If G is a finite extension of a soluble group with finite abelian ranks, then G is abelian-by-finite.     

The evaluation of Gauss sums for characters of 2-power order

Let χ be a character of order 2^t on the splitting field of the polynomial x^2t−1 over the prime field with p≡3,5 (mod 8). In this paper we evaluate explicitly the Gauss sums related to the group of characters generated by χ.     

Cox rings of K3 surfaces with Picard number 2

We study presentations of Cox rings of K3 surfaces of Picard number 2. In particular we consider the Cox rings of classical examples of K3 surfaces, such as quartic surfaces containing a line and doubly elliptic K3 surfaces.     

Bounded automorphisms of surfaces

The equivalence among bounded homeomorphisms of a surface, reducible homeomorphisms of the surface, bounded automorphisms of the surface group, induced or exchange automorphisms of the surface group, and the stabilizer of the translation length function of the tree action is established.     

Hadamard matrices of order 28 with automorphisms of order 7

The only primes which can divide the order of the automorphism group of a Hadamard matrix of order 28 are 13, 7, 3, and 2, and there are precisely four inequivalent matrices with automorphisms of order 13 (Tonchev, J. Combin. Theory Ser. A35 (1983), 43–57). In this paper we show that there are exactly twelve inequivalent Hadamard matrices of order 28 with automorphisms of order 7. In particular, there are precisely seven matrices with transitive automorphism groups.     

Groups with regular automorphisms of order four

Mathematische Zeitschrift -     

Polycyclic groups with automorphisms of order four

n this paper, we study the structure of polycyclic groups admitting an automorphism of order four on the basis of Neumann’s result, and prove that if α is an automorphism of order four of a polycyclic group G and the map φ: G → G defined by g^φ = [g,α] is surjective, then G contains a characteristic subgroup H of finite index such that the second derived subgroup H″ is included in the centre of H and C_H(α²) is abelian, both C_G(α²) and G/[G, α²] are abelian-by-finite. These results extend recent and classical results in the literature.     

Uniformization of surfaces of genus two with automorphisms

Research partially supported by the United Nations Development Program CHI-84004