2 Supersingular K3 surfaces

A K3 surface over an algebraically closed field $k=\overline{k}$ is a smooth surface with

Common examples are smooth quartics in $\mathbb{P}^{3}$ and double covers of $\mathbb{P}^{2}$ branched over a smooth sextic. The group of divisors modulo algebraic equivalence is called the Néron–Severi group $NS(X)$ . Its rank is called the Picard number. Equipped with the intersection pairing, $NS(X)$ is a lattice (see Section 4). A singular K3 surface over $\mathbb{C}$ is one whose Picard number

which is the maximal possible. Here “singular” is meant in the sense of exceptional rather than nonsmooth. In positive characteristic, however, the Picard number is only bounded by the second Betti number $b_{2}(X)=22$ , and K3 surfaces reaching the maximum possible Picard number

are called supersingular. Supersingular K3 surfaces are classified according to their Artin invariant $\unicode[STIX]{x1D70E}$ defined by $\det NS(X)=-p^{2\unicode[STIX]{x1D70E}},\unicode[STIX]{x1D70E}\in \{1,\ldots ,10\}$ (see [Reference Artin1]). By the work of Ogus [Reference Ogus10], there is a unique K3 surface of Artin invariant $\unicode[STIX]{x1D70E}=1$ , over $k=\overline{k}$ of characteristic $p$ . We denote it by $X(p)$ .

Supersingular K3 surfaces arise from singular K3 surfaces as follows.

As noticed by Shimada [Reference Shimada14], if $\det NS(X)$ is coprime to $p$ , then the Artin invariant is $\unicode[STIX]{x1D70E}=1$ . The reason for this is that $NS(X){\hookrightarrow}NS(X_{p})$ implies $\unicode[STIX]{x1D70E}=1$ .

3 Automorphisms and dynamics

A Salem polynomial is an irreducible monic polynomial $S(x)\in \mathbb{Z}[x]$ of degree $2d$ such that its complex roots are of the form

The unique positive root $a$ outside the unit disk is called a Salem number.

The following theorem is due to McMullen [Reference McMullen9] over $\mathbb{C}$ and Esnault and Srinivas [Reference Esnault and Srinivas5] in positive characteristic.

Notice that over $\mathbb{C}$ we can work equally well with the singular cohomology groups $H^{2}(X,\mathbb{Z})$ and prove the result via Hodge decomposition. The theorem motivates the following definition. Given a smooth projective surface over an algebraically closed field and an automorphism $f:X\rightarrow X$ , the entropy $h(f)$ of $f$ is defined as

where $r(f^{\ast }|NS(X))$ is the spectral radius of the linear map $f^{\ast }|NS(X)\otimes \mathbb{Q}$ ; that is, the maximum of the absolute values of its complex eigenvalues. The entropy is either zero or the logarithm of a Salem number $a$ . The degree of the Salem polynomial corresponding to $a$ is called the Salem degree of  $f$ .

By the work of Esnault and Srinivas [Reference Esnault and Srinivas5], this is compatible with the definition of topological entropy for smooth complex projective surfaces.

Over $\mathbb{C}$ , due to Hodge theory, the rank of the Néron–Severi group is bounded by $20=h^{1,1}(X)$ . Hence, an automorphism on a complex projective K3 surface has Salem degree at most 20. In the nonprojective case, McMullen [Reference McMullen9] constructed K3 surfaces with automorphisms of Salem degree 22. In positive characteristic, however, K3 surfaces may have Picard rank 22, and there automorphisms of Salem degree 22 occur. A specific feature of such automorphisms is that they do not lift to any characteristic zero model.

4 Lattices

In this section, we fix our notation concerning lattices. A lattice $L$ is a finitely generated free $\mathbb{Z}$ module equipped with a nondegenerate symmetric bilinear pairing $L\times L\rightarrow \mathbb{Q}$ . If it is integer valued, then the lattice is called integral. The pairing identifies

If $L$ is integral, it induces an embedding $L{\hookrightarrow}L^{\ast }$ . The finite quotient $A_{L}:=L^{\ast }/L$ is called the discriminant group of $L$ . Given a $\mathbb{Z}$ -basis $(b_{i})$ , the determinant of the Gram matrix $(b_{i}.b_{j})_{ij}$ is independent of the choice of basis and is called the determinant of  $L$ . Its absolute value is the order of the discriminant group. In this basis, the dual lattice is generated by the columns of $(b_{ij})^{-1}$ . Given an embedding $M{\hookrightarrow}L$ of integral lattices of the same rank, $A_{L}$ is a subquotient of $A_{M}$ and we get

A lattice is called unimodular if it is integral and has $|\text{det}\,L|=1$ . An embedding $M{\hookrightarrow}L$ of lattices is called primitive if $L/M:=L/im(M)$ is torsion free. If $M{\hookrightarrow}L$ is a primitive embedding of integral lattices and $L$ is unimodular, then $A_{M}\cong A_{M^{\bot }}$ , where $M^{\bot }$ is the orthogonal complement of $M$ in $L$ . We define the length of a finite abelian group by its minimum number of generators. Note that $l(A_{L})\leqslant \text{rk}\,L$ . A lattice is called $p$ -elementary if $pA_{L}=0$ , or equivalently if $A_{L}$ is an $\mathbb{F}_{p}$ vector space.

5 Elliptic fibrations on K3 surfaces

A genus-one fibration on a surface $X$ is a surjective map

to a smooth curve $C$ such that the generic fiber is a smooth curve of genus one. We call it an elliptic fibration, if the additional datum of a section $O$ of $\unicode[STIX]{x1D70B}$ is given. Indeed, all genus-one fibrations occurring throughout this paper admit a section. This turns the generic fiber of an elliptic fibration into an elliptic curve $E$ over $K=k(C)$ , the function field of $C$ . For a K3 surface, $C=\mathbb{P}^{1}$ is the only possibility. There is a one to one correspondence between $K$ -rational points of $E$ and sections of $\unicode[STIX]{x1D70B}$ . Both of these sets are abelian groups, which we call the Mordell–Weil group of the fibration. This is denoted by $\text{MW}(X,\unicode[STIX]{x1D70B},O)$ , where $\unicode[STIX]{x1D70B}$ and $O$ are suppressed from the notation if confusion is unlikely. The addition on $\text{MW}$ is denoted by  $\oplus$ .

A good part of $NS$ is readily available: the trivial lattice

If $O$ and $\unicode[STIX]{x1D70B}$ are understood, we suppress them from the notation. By the results of Kodaira [Reference Kodaira7] and Tate [Reference Tate17], the trivial lattice decomposes as an orthogonal direct sum of a hyperbolic plane spanned by $O$ together with the fiber $F$ and negative definite root lattices of type $ADE$ consisting of fiber components disjoint from $O$ . Note that the singular fibers (except in some cases in characteristics 2 and 3) are determined by the $j$ -invariant and discriminant of the elliptic curve  $E$ .

An advantage of elliptic fibrations is that they structure the Néron–Severi group into sections and fibers, as given by the following theorem.

The Mordell–Weil group can be equipped with a positive definite symmetric $\mathbb{Q}$ -valued bilinear form via the orthogonal projection with respect to $\text{Triv}(X)$ in $NS(X)\otimes \mathbb{Q}$ and switching sign. Explicitly, it is given as follows. Let $P,Q\in \text{MW}(X,O)$ , and denote by $R$ the set of singular fibers of the fibration. Then,

where the dot denotes the intersection product on the smooth surface $X$ . The term $c_{\unicode[STIX]{x1D708}}(P,Q)$ is the local contribution at a singular fiber, given as follows. If one of the two sections involved meets the same component of the singular fiber $\unicode[STIX]{x1D708}$ as the zero section, then the contribution at $\unicode[STIX]{x1D708}$ is zero. If this is not the case, the contribution is nonzero and depends on the fiber type. We only need the types  $\mathit{III}$ , $\mathit{III}^{\ast }$ and  $I_{0}^{\ast }$ ; for the others consult [Reference Shioda13]. If $\unicode[STIX]{x1D708}$ is of type  $\mathit{III}^{\ast }$ (resp.  $\mathit{III}$ ), the contribution is equal to $3/2$ (resp.  $1/2$ ) if $P$ and $Q$ meet the same component of $\unicode[STIX]{x1D708}$ and zero otherwise. For $\unicode[STIX]{x1D708}$ of type  $I_{0}^{\ast }$ , we have $c_{\unicode[STIX]{x1D708}}(P,Q)=1$ if they meet in the same component and $1/2$ otherwise. Equipped with this pairing, $\text{MWL}(X):=\text{MW}(X)/\text{torsion}$ is a positive definite lattice. In general, it is not integral.

6 Isotrivial fibration

In this section, we compute generators of the Néron–Severi group as well as their intersection matrix.

Note that the $j$ -invariant $j=1728$ is constant. Hence, all smooth fibers are isomorphic – such a fibration is called isotrivial.

Our next task is to work out generators of $NS(X(p))$ for $p\equiv 3~\text{mod}\,4$ . By Theorem 5.1, it is generated by sections and fiber components. Reducing $j$ and $\unicode[STIX]{x1D6E5}~\text{mod}\,p$ , we see that the fiber types are preserved mod $p$ (even in case $p=3$ ; cf. [Reference Silverman16, p. 365]). Hence, sections of infinite order must appear. Generally, it is hard to compute sections of an elliptic fibration. For this special fibration, there is a trick involving a purely inseparable base change of degree $p$ turning $X(p)$ into a Zariski surface.

For later reference, we fix the following $\mathbb{Z}$ -basis of the Néron–Severi group, where the fiber components are sorted as indicated in Figure 1, and $e_{20}$ is distinguished by $e_{20}.P=1$ :

Blue vertexes and edges belong to fibers. Replacing $Q$ by one of the missing components of the $I_{0}^{\ast }$ fiber results in a $\mathbb{Q}$ - instead of a $\mathbb{Z}$ -basis. This is predicted by Theorem 5.1.

Figure 1.

Twenty-four $(-2)$ -curves of $X$ supporting singular fibers of type $I_{0}^{\ast },2\times \mathit{III}^{\ast }$ (blue) and torsion sections of  $\unicode[STIX]{x1D70B}$ .

For the intersection matrix in this basis, one obtains

In the remaining part of this section, we explain how the sections $P,R$ are found and give an alternative way of computing their intersection numbers.

Recall that we assume that $p\equiv 3~\text{mod}\,4$ and write $p=4n+3$ . Consider the purely inseparable base change $t\mapsto t^{p}$ . This changes the equation as follows:

We minimize this equation using the birational map

This leads to the surface $Y$ given by

After another base change $t\mapsto t^{p}$ , we get

and minimizing the fibration

we recover $X$ .

Instead of directly searching for sections on $X$ , we exhibit two sections on $Y$ and pull them back to $X$ . The $j$ -invariant of $Y$ is still equal to 1728, but $\unicode[STIX]{x1D6E5}=-2^{6}t^{3}(t-1)^{6}$ . For $p\neq 3$ , this leads to two singular fibers of type $\mathit{III}$ over $t=0,\infty$ and a singular fiber of type $I_{0}^{\ast }$ over $t=1$ . The Euler number of this surface is $2e(\mathit{III})+e(I_{0}^{\ast })=2\cdot 3+6=12$ . By general theory, it is rational. The trivial lattice is isomorphic to

This lattice is of determinant $-16$ and rank 8. From the theory of elliptically fibered rational surfaces, we know that $\text{rk}\,NS(Y)=10$ , $\det NS(Y)=-1$ , which implies that $Y$ has Mordell–Weil rank 2. Define

Then, by Theorem 5.1, $\text{Triv}^{\prime }(Y)/\text{Triv}(Y)\cong \text{MW}_{tors}$ . Since $\{x=y=0\}$ is a 2-torsion section, we know that

As even unimodular lattices of signature $(1,7)$ do not exist, $-1$ is impossible. We conclude that $x=y=0$ is the only torsion section.

To find a section of infinite order, we first determine the Mordell–Weil lattice and then translate this information to equations of the section. Since $j=1728$ , the elliptic curve admits complex multiplication given by $(x,y)\mapsto (-x,iy)$ . Obviously, $Q$ and $O$ are the unique sections fixed by this action. Hence, the Mordell–Weil lattice admits an isometry of order four, which, viewed as an element of $O(2)$ , may only be a rotation by $\pm {\textstyle \frac{\unicode[STIX]{x1D70B}}{2}}$ . Up to isometry, all positive definite lattices of rank 2 admitting an isometry of order four have a Gram matrix of the form

for some $a>0$ . The formula

where $r=rk\,\text{MWL}(Y)=2$ , $\det NS(Y)=-1$ , $\det \text{Triv}^{\prime }(Y)=4$ , resulting from the definition of $\text{MWL}(Y)$ via the orthogonal projection with respect to $\text{Triv}^{\prime }(Y)$ , yields $a=1/2$ .

We search for a section $P$ with

where $\unicode[STIX]{x1D712}=\unicode[STIX]{x1D712}({\mathcal{O}}_{Y})=1$ is the Euler characteristic of $Y$ and $c_{0},c_{\infty }\in \{0,1/2\},c_{1}\in \{0,1\}$ are the contributions at the singular fibers; $P.O\in \mathbb{N}$ is the intersection number. This immediately implies $P.O=0$ and $c_{1}=1$ . Let us assume $c_{0}=1/2,c_{\infty }=0$ . The section $P$ can be given by $(x(t),y(t))$ , where $x,y$ are rational functions. Over the chart containing $\infty$ , it is given by $(\hat{x}(s),{\hat{y}}(s))=(s^{2}x(1/s),s^{3}y(1/s))$ , where $s=1/t$ . As poles of these functions correspond to intersections with the zero section, we know that $x,y,\hat{x},{\hat{y}}$ are actually polynomials. Therefore, $\deg x\leqslant 2$ and $\deg y\leqslant 3$ . Furthermore, if $P$ intersects the same fiber component as the zero section, then the contribution $c_{\unicode[STIX]{x1D708}}$ of that fiber is zero. The other components arise by blowing up the singularities of the Weierstraß model in $\{x=y=t(t-1)=0\}$ . Hence, $x(0)=y(0)=x(1)=y(1)=0$ . Putting this information together, we obtain $x=at(t-1)$ and $y=t(t-1)b$ for some constant $a$ and a polynomial $b(x)$ of degree one. A quick calculation yields the sections

for $\unicode[STIX]{x1D701}^{4}+1=0$ . Note that $\text{MWL}(Y)$ contains exactly four sections of height $1/2$ . Furthermore, $\langle P^{\prime },R^{\prime }\rangle =0$ . (Otherwise, $P=\ominus R$ , which is clearly false.) The two further Galois conjugates of $\unicode[STIX]{x1D701}$ correspond to the missing two sections $\ominus P$ and $\ominus R$ .

Now we pull back $P^{\prime }$ and $R^{\prime }$ to $X$ via the map

and get the sections

It remains to compute the intersection numbers involving $P$ and $R$ . This can be done by blowing up the singularities and then computing the intersections. However, by applying some more of Shioda’s theory, we can avoid the blowing up. From the behavior of the height pairing under base extension (cf. [Reference Shioda13, Proposition 8.12]), we know that

and get the equation

Note that a fiber of type $\mathit{III}^{\ast }$ has only two simple components. Since $P$ meets the same component over infinity as the zero section and passes through the singularities at $t=0,1$ , we know that $c_{\infty }=0$ , $c_{0}=3/2$ and $c_{1}=1$ . We conclude that $P.O=n$ . The other intersection numbers can be calculated accordingly. In this way, we could calculate the intersection matrix of $NS(X(p))$ without knowing equations for the extra sections.

The section $P$ induces an automorphism of the surface by fiberwise addition. We denote it by $(\oplus P)$ . The matrix of its representation on $NS$ is obtained as follows.

• ∙ Compute the basis representation of the sections $Q\oplus P,2P$ and $R\oplus P$ .

• ∙ Any section $S$ is mapped to $P\oplus S$ under $(\oplus P)_{\ast }$ , and the fiber is fixed.

• ∙ The action of $(\oplus P)$ on the Néron–Severi group preserves each singular fiber. Since it is an isometry, it can be determined by its action on sections.

• ∙ Invert the resulting matrix to get $(\oplus P)^{\ast }=(\oplus P)_{\ast }^{-1}$ .

A basis representation of $P\oplus Q$ is obtained as follows. Start with $P+Q\in NS$ and subtract $nO$ , such that the resulting divisor $D$ has $D.F=1$ . Add or subtract fiber components until $D$ meets each singular fiber in exactly one simple component. Finally, add multiples of $F$ such that $D^{2}=-2$ . For example, the basis representation of the section $P\oplus R$ is given by

7 Alternative elliptic fibrations

The automorphism constructed in the last section has zero entropy. The reason for this is that it fixes the fibers. We overcome this obstruction by combining different fibrations and their sections.

Reducible singular fibers of elliptic fibrations are extended $ADE$ -configurations of $(-2)$ -curves. Conversely, let $X$ be a K3 surface, and let $F$ be an extended $ADE$ -configuration of $(-2)$ -curves. Then, the linear system $|F|$ is an elliptic pencil with the extended $ADE$ -configuration as singular fiber. See [Reference Kondō8] and [Reference Pjateckiĭ-Šapiro and Šafarevič11] for details. We use this fact to detect additional elliptic fibrations in the graph of $(-2)$ -curves. Irreducible curves $C$ are either fiber components, sections or multisections, depending on whether $C.F=0,1$ or ${>}1$ . Be aware that the $(-2)$ -curves occur with multiplicities in $F$ . Hence, it is possible that $C.F>1$ even if $C$ meets only a single $(-2)$ -curve of the configuration. In general, an elliptic pencil does not necessarily admit a section. However, once its existence is guaranteed, we may choose it as zero section. Then, by Theorem 5.1, multisections still induce sections once modified by fiber components and the zero section, as sketched above.

Figure 2.

$\unicode[STIX]{x1D70B}^{\prime }$ with $I_{16}$ and $I_{4}$ fibers and $\unicode[STIX]{x1D70B}^{\prime \prime }$ with $I_{12}$ and $IV^{\ast }$ fibers.

The first fibration $\unicode[STIX]{x1D70B}^{\prime }$ is induced by the outer circle of $(-2)$ -curves, which is a singular fiber of type $I_{16}$ . There is a second singular fiber of type $I_{4}$ . Three of its components are visible in Figure 2. The curve $e_{8}$ ( $=$ vertex labeled by “8”) is a section since it meets $I_{16}$ exactly in a simple component. We take it as zero section. Then, the torsion sections are $e_{15},e_{18}$ and $e_{19}$ . The second fibration $\unicode[STIX]{x1D70B}^{\prime \prime }$ is induced by the right inner circle of $(-2)$ -curves. It has singular fibers of type $I_{12}$ and $IV^{\ast }$ , and again we take $e_{8}$ as zero section. A simple component of the $IV^{\ast }$ fiber is not visible in Figure 2. In both cases, $P$ is a multisection and induces a section of each fibration denoted by $P^{\prime }$ and $P^{\prime \prime }$ . For example, the class of $P^{\prime }\in NS(X(p))$ is given by

8 Salem degree-22 automorphism

We consider the automorphism

on $X(p)$ for $p\equiv 3~\text{mod}\,4$ . Using a computer algebra system, one computes the characteristic polynomial of $f^{\ast }|NS(X(p))$ :

where

By Theorem 3.1, $\unicode[STIX]{x1D707}(f^{\ast })$ either is a degree-22 Salem polynomial or is divisible by a cyclotomic polynomial of degree at most 22. There are only finitely many cyclotomic polynomials of a given degree. We can exclude the second case directly by computing the remainder after division for each such polynomial. This proves Theorem 1.1.