Orders of automorphisms of K3 surfaces

Abstract: We determine all posible orders of automorphisms of finite order of complex K3 surfaces or of K3 surfaces in characteristic $p>3$. E.g., a positive integer $N$ is the order of an automorphism of a complex K3 surface if and only if $φ(N)\le 20$ where $φ$ is the Euler function. In particular, 66 is the maximum finite order in each characteristic $p\neq 2,3$. As a consequence, we give a bound for the orders of finite groups acting on K3 surfaces in characteristic $p>7$.

• The paper provides a complete classification of finite automorphism orders on K3 surfaces by demonstrating that Euler’s totient function constraint (φ(N) ≤ 20) governs realizable orders.
• It uses a cohomological approach to analyze eigenvalue structures on H², constructing explicit examples in various characteristics and resolving long-standing exceptions.
• The work clarifies the behavior of tame and wild automorphisms in positive characteristics, connecting surface geometry, group theory, and arithmetic invariants.

Determination of Automorphism Orders of K3 Surfaces in Arbitrary Characteristic

Introduction and Problem Overview

The paper "Orders of automorphisms of K3 surfaces" (1203.5616) offers a complete and rigorous classification of possible finite orders of automorphisms of K3 surfaces over algebraically closed fields, with a particular focus on separating the characteristic $p$ of the base field. The significance stems from the profound interplay between surface geometry, arithmetic, and the cohomological properties encoded in these symmetries. Precise knowledge of automorphism orders has direct implications for the structure of the automorphism group $\operatorname{Aut}(X)$, the geometry of moduli spaces, and connections with arithmetic cohomology and algebraic groups.

Prior classifications had only covered symplectic and purely non-symplectic automorphisms for complex K3 surfaces, but no general result for all finite automorphisms, especially in positive characteristic, was available. This work provides a general solution—except for characteristics 2 and 3—by unifying case analysis via cohomological invariants and constructing explicit examples in each case.

Main Results

Complete List of Orders

A central theorem established is that for complex K3 surfaces or K3 surfaces in characteristic $p>3$, the set $\operatorname{Ord}_p$ of possible finite automorphism orders is: $$\operatorname{Ord}_\mathbb{C} = \{N \in \mathbb{N}^+ \mid \varphi(N) \leq 20\}$$ with bounded exceptions for positive characteristic, especially for small $p$. $\varphi$ denotes Euler's totient function. In particular, the maximum finite automorphism order is achieved for $N = 66$, with

$\varphi(66) = 20$

This order is realized both in characteristic zero and in most positive characteristics $p \neq 2,3$ (with explicit adjustments for small primes).

For $p=7$ or $p>19$, $$\operatorname{Ord}_p = \operatorname{Ord}_\mathbb{C} \setminus \{p,2p\}$$; for $p=11$, $13$, $17$, $19$, or $5$, further exceptions occur, dictated by the possible degeneracies in the structure of wild automorphisms and explicit pathological cases.

For $p=2$ or $3$, the result is more intricate due to wild automorphisms, but the criterion for tame automorphisms (those of order coprime to $p$) remains $\varphi(N) \leq 20$.

This result subsumes, sharpens, and extends bounds previously only known in special cases, such as the upper bound $66$ for complex K3 surfaces (contrasting with the traditional bound $528$ derived as the product of $8$ for symplectic and $66$ for purely non-symplectic automorphisms).

Methodology

The classification leverages a cohomological approach: the action of an automorphism $g$ on the second (étale/singular) cohomology group is analyzed via its eigenvalues. The key observation is that if $g$ has order $N$ and acts on the Néron-Severi group and the transcendental part, the eigenvalues are necessarily roots of unity of order dividing $N$, and $\varphi(N)$ bounds the possible dimension of eigenspaces. The dimension of $H^2$ is $22$ for K3 surfaces, thereby generating the upper bound on $\varphi(N)$.

Furthermore, significant technical extensions are required in positive characteristic: Weil/Deligne cohomology replaces classic topological methods, and special arguments handle the failure of the Lefschetz fixed point formula in wild characteristics (when order is divisible by $p$). Sharp case analyses and explicit examples are used to establish the realization of each possible case.

Faithful representations of $\operatorname{Aut}(X)$ on cohomology (Ogus's theorem) play a central role in controlling which orders can be realized.

Examples and Explicit Constructions

For every admissible $N$, the paper constructs explicit K3 surfaces (in characteristic $0$ and $p > 3$) admitting an automorphism of order $N$. These constructions are typically elliptic K3 surfaces or double covers with suitable Weierstrass or branched equations, ensuring good reduction modulo $p$ for $p \nmid N$. For pathological characteristics ($p = 2,3$), special examples are constructed demonstrating the compatibility and exceptions in the theory.

Orders of "Wild" Automorphisms

Wild automorphisms (orders divisible by $p$) only occur for $p \leq 11$. Their possible orders in the case $p = 5,7,11$ are explicitly determined through detailed cohomological and geometric analysis, leveraging results of Deligne-Lusztig, group theory, and geometry of fixed loci.

Group Actions and Bounds

A further corollary addresses the maximal possible orders of finite groups $G$ acting as automorphisms on a K3 surface. In characteristic $0$, the maximal group order is bounded by $3840$ (complex case, Kondō), and $960$ for symplectic actions (Mukai). In positive characteristic, much larger group actions can occur for wild automorphisms, e.g., PSU$_3(5)$ of order $126,000$ and Mathieu group $M_{22}$ of order $443,520$. However, the main theorem demonstrates that, except for $p = 2, 3$, the set of possible automorphism orders does not expand in positive characteristic.

Theoretical and Practical Implications

This classification resolves an outstanding algebraic and arithmetic question for K3 surfaces, with multiple implications:

• Moduli Theory: The result constrains possible stack structures of moduli of K3 surfaces with level structures and sheds light on the possible degenerations.
• Group Theory and Arithmetic Geometry: The results connect K3 automorphism groups with finite simple and sporadic groups and inform about possible symmetries in low characteristic.
• Cohomological Invariants: The proof techniques further the interplay between algebraic cycles, crystalline/étale cohomology, and autoequivalences of derived categories in positive characteristic.
• Comparison with Abelian Surfaces and Enriques Surfaces: The results illustrate both the parallels and essential differences in automorphism behavior among higher genus surfaces (see remarks comparing to elliptic and abelian surfaces).

Future work may seek analogous complete classifications in characteristics $2$ and $3$, explore the enumerative geometry of K3 moduli under group actions, and refine the relationship between automorphism order, the Picard rank, and the geometry of supersingular K3s, as well as potential links to arithmetic dynamics and derived autoequivalences.

Conclusion

This work rigorously establishes that the set of possible finite orders of automorphisms of K3 surfaces is determined by the Euler function constraint $\varphi(N) \leq 20$, with maximal order $66$, barring explicit and exceptional scenarios in small characteristic. The result holds with explicit examples demonstrating realizability and proves that new automorphism orders peculiar to positive characteristic ($p > 3$) do not arise for tame automorphisms. This brings a long-standing question to a complete solution for all characteristics except $2$ and $3$, and offers a template for similar systematic investigations for other surfaces of general type and higher-dimensional Calabi-Yau varieties.