Some Remarks on Non-Symplectic Automorphisms of K3 Surfaces Over a Field of Odd Characteristic

Abstract

Abstract. In this paper, we present a simple proof of Corollary 3.3 in [5] using the fact that for a K3 surface of finite height over a field of odd characteristic, the height is a multiple of the non-symplectic order. Also we prove for a non-symplectic CM K3 surface defined over a number field, the Frobenius invariant of the reduction over a finite field is determined by the congruence class of residue characteristic modulo the non-symplectic order of the K3 surface.

The Non-symplectic Index of Supersingular K3 Surfaces

“…For such a υ, the height (and the Artin invariant) of X υ is determined by the congruence class of the residue characteristic p υ modulo N . In particular, if m is the smallest positive integer such that p m υ ≡ −1 modulo N , then X υ is a supersingular of Artin invariant m. ( [8], Theorem 4.7, [6], Theorem 2.3) Moreover, in this case, ν Xυ ∈ O(l(N S(X υ ))) has 2σ distinct eigenvalues, so a i = 0 for all i in (2.1) and X υ is a special supersingular K3 surface.…”

On the supersingular reduction of K3 surfaces with complex multiplication

“…unity and φ is Euler's totient function. For such a K3 surface X C , Theorem 1.1 and Theorem 1.2 were proved by Jang when p = 2; see [15,Theorem 2.3], [16,Corollary 3.3].…”