---
abstract: 'Consider an arbitrary automorphism of an Enriques surface with its lift to the covering $K3$ surface. We prove a bound of the order of the lift acting on the anti-invariant cohomology sublattice of the Enriques involution. We use it to obtain some mod 2 constraint on the original automorphism. As an application, we give a necessary condition for Salem numbers to be dynamical degrees on Enriques surfaces and obtain a new lower bound on the minimal value. In the Appendix, we give a complete list of Salem numbers that potentially may be the minimal dynamical degree on Enriques surfaces and for which the existence of geometric automorphisms is unknown.'
address:
- 'Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, 464-8602, Japan'
- 'Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science, 2641 Yamazaki, Noda, Chiba 278-8510, Japan'
- 'Institute of Mathematics, Jagiellonian University, ul. [Ł]{}ojasiewicza 6, 30-348 Kraków, Poland'
author:
- Yuya Matsumoto
- Hisanori Ohashi
- 'S[Ł]{}awomir Rams'
title: On automorphisms of Enriques surfaces and their entropy
---

[^1]

introduction
============

It is known that the only compact Kähler surfaces that admit automorphisms of positive topological entropy are rational, Enriques, $K3$ surfaces and complex tori (see e.g. [@cantat $\S$ 2.5]). Salem numbers that can be realized as the dynamical degrees of automorphisms of 2-dimensional tori are fully characterized in [@reschke12 Thm 1.1] in terms of values of the minimal polynomials. The same question is solved for rational surfaces in terms of Weyl groups in [@Uehara]. These are exactly the description of the dynamical spectrum $$\Lambda(\mathcal{C})=\{\lambda(f)\in\mathbb{C}\mid \lambda(f)\text{is the dynamical degree of $f\in \mathrm{Aut}(S)$ for some $S\in \mathcal{C}$}\}$$ where the class of surfaces $\mathcal{C}$ is taken to be $2$-tori or rational surfaces. For $K3$ surfaces, the recent preprint [@BF-T] describes the case of degree 22 Salem numbers. Other degrees on $K3$ surfaces and also on Enriques surfaces the description of $\Lambda(\mathcal{C})$ remains open.

The purpose of this note is to give a new property which is satisfied by all automorphisms of Enriques surfaces. As a consequence, we obtain a new constraint on the Salem numbers that appear as the dynamical degrees of automorphisms of Enriques surfaces, namely a property of $\Lambda
(\text{Enriques})$. It should be noted that despite its ergodic interpretation [@cantat $\S$.2.2.2], the problem we consider is purely algebraic, in the sense that the dynamical degree of an automorphism of an Enriques surface $S$ can be detected as the spectral radius of the map it induces on $\mathrm{Num}(S)$ (see $\S$.\[sect-dynamical-degrees\]).

To state the theorem, let $S$ be an Enriques surface and let $\tilde{S}$ be its $K3$-cover. We denote by $\varepsilon$ the covering involution of the double étale cover $\pi: \tilde{S} \rightarrow S$ and put $N$ to denote the orthogonal complement of the $\varepsilon$-invariant sublattice $H^2(\tilde{S},\Z)^{\varepsilon}$ in the lattice $H^2(\tilde{S},\Z)$: $$\label{eq-latN}
N = (H^2(\tilde{S},\Z)^{\varepsilon})^{\perp} \, .$$ Recall that for an arbitrary automorphism $f\in \mathrm{Aut}(S)$, there exists a lift $\tilde{f} \in \mathrm{Aut}(\tilde{S})$. Obviously the lift in question is not unique (given $\tilde{f}$, the automorphism $\tilde{f} \circ \varepsilon$ is also a lift of $f$), but the constraints we prove are valid for any choice of $\tilde{f}$. As is well-known (see e.g. [**[@namikawa]**]{}), the lattice $N$ is stable under the cohomological action $\tilde{f}^*$, hence the restriction $${f_N}:= \tilde f^* \rvert_N$$ is an automorphism (isometry) of $N$. It is easy to see that the order $\ord({f_N})$ is finite, Lemma \[Nfin\]. Here we show a more precise constraint on the order of the map ${f_N}$:

\[th1\] Let $S$ be an Enriques surface and let $f\in \mathrm{Aut}(S)$. Then, the order of ${f_N}$ is an integer which divides at least one of the integers $$120, 90, 84, 72, 56, 48.$$ Equivalently and explicitly, these are one of the 31 integers $$\label{eq-T1}
120,90,84,72,60,56,
 48,45,42,40,36,30,28, 24,21,20,18, 16, 15,14,12,10,\dots,1.$$

We use the above theorem to derive the following mod 2 constraint for a Salem number to be the dynamical degree of automorphisms of Enriques surfaces (which refines [@third Lemma 4.1]):

\[mod2\] Let $f$ be an automorphism of an Enriques surface $S$ and let $s_{\lambda}$ be the minimal polynomial of its dynamical degree $\lambda(f)$. Then the modulo $2$ reduction of $s_{\lambda}$ is a product of (some of) the following polynomials $$\begin{aligned}
F_{ 1}(x) &=                                                          x + 1, \quad 
F_{ 3}(x)  =                                                    x^2 + x + 1, \quad 
F_{ 5}(x)  =                                        x^4 + x^3 + x^2 + x + 1, \\
F_{ 7}(x) &=                            x^6 + x^5 + x^4 + x^3 + x^2 + x + 1, \\
F_{ 9}(x) &=                            x^6 +             x^3           + 1, \\
F_{15}(x) &=                x^8 + x^7 +       x^5 + x^4 + x^3 +       x + 1, \\\end{aligned}$$

Here each $F_m(x) \in \F_2[x]$ is the modulo $2$ reduction of the $m$-th cyclotomic polynomial $\Phi_m(x)\in \mathbb{Z}[x]$. Among these six polynomials, $F_7(x)$ and $F_{15}(x)$ are products of two distinct irreducible factors, each of which is not self-reciprocal, whereas the other four are irreducible.

We do not know whether all the six factors above do appear among factorizations of the minimal polynomial of the map induced by an automorphism of Enriques surfaces, but we are able to give examples where $F_1(x), F_3(x), F_5(x)$ do come up (see Example \[example-existence-3factors\].a). On the other hand, we can check that they all appear from some [*[lattice isometry]{}*]{} of $U\oplus E_8$ by using lattice theory and ATLAS table, for example.

A closely related problem to the description of the dynamical spectrum is to find the minimal nontrivial dynamical degree $1\neq \lambda\in \Lambda(\mathcal{C})$.

This question was answered for complex tori (see [@mcmullen11 Thm 1.3]), rational and $K3$ surfaces by McMullen (see [@mcmullen07; @mcmullen11; @mcmullen16]), whereas the smallest dynamical degree attained by automorphisms of Enriques surfaces is yet to be found ([@third Question 4.5.(3)]). By [@third Remark 4.4], none of the smallest five Salem numbers (including the ones of degree $>10$) can be realized on Enriques surfaces. Although explicit descriptions of automorphism groups of several special families (see [@BP; @MO15]) of Enriques surfaces are known, our present knowledge seems not to be enough to determine the minimal dynamical degree of automorphisms of surfaces in this class. Presently the smallest known dynamical degree of an automorphism of an Enriques surface is the one constructed by Dolgachev [@dolgachev16 Table 2], who found an automorphism of an Enriques surface (of Hesse type) of dynamical degree $\lambda_D=2.08101\ldots$ (see Example \[Dolgachevexample\] for more details). As to this respect, in Example \[MO15family\], we give an additional study on the family of [@MO15] to show that all nontrivial dynamical degrees of automorphisms of Enriques surfaces in [@MO15] are at least $\lambda_D$. The result thus fails to give a new lower bound, but gives a good account for what is going on.

To fill the gap, the constraint given by Thm \[mod2\] works to give the following slightly better theoretical lower bound:

\[th2\] The dynamical degree of an automorphism of an Enriques surface is greater than or equal to the Salem number $\lambda=1.35098\cdots$ given by the polynomial $$x^{10}-x^9-x^6+x^5-x^4-x+1.$$

We use Theorem \[mod2\] together with [@Gross--McMullen], combined with Dolgachev’s example [@dolgachev16 Table 2] to show that the smallest (non-trivial) dynamical degree of an automorphism of an Enriques surface must be one of the 39 Salem numbers which we list in $\S.$\[sect-appendix\] Appendix.

Our approach is inspired by Oguiso’s proof of [@third Thm 1.2]. We consider mod 2 reduction of the cohomological action, and apply results from [@Gross--McMullen] to obtain a detailed picture. It should be noted that our approach does rule out numerous Salem numbers (see Example \[rem-possibilities\]), but Thm \[mod2\] cannot lead to a necessary and sufficient condition for a Salem number to be the dynamical degree of an automorphism of an Enriques surface. One possible way of determining the exact minimal value of non-trivial dynamical degrees of automorphisms of Enriques surfaces will be that the remaining $39$ cases can be treated efficiently by some refinement of McMullen’s method [@mcmullen16], but this task exceeds the scope of this paper.

[**Convention:**]{} In this note we work over the field of complex numbers $\C$.

Proof of Theorem \[th1\] {#sect-th1}
========================

We maintain the notation of the introduction: $S$ is assumed to be an Enriques surface and $f\in \mathrm{Aut}(S)$ is an automorphism. As is well-known, the canonical cover $\tilde{S}=\mathrm{Spec}(\mathcal{O}\oplus \mathcal{O}(K_S))$ is a $K3$ surface and the morphism $\pi: \tilde{S} \rightarrow S$ is a double étale cover. We denote by $\varepsilon$ the covering involution of $\pi$. Since the automorphism $f$ preserves the canonical class $K_S\in \mathrm{Pic}(S)$, $f$ lifts to an automorphism $\tilde{f}$ of $\tilde{S}$.

Recall that for an Enriques surface the lattice $\mathrm{Num}(S)$ is the free part of the cohomology group $H^2(S,\Z)$. Let $$M:=H^2(\tilde{S},\Z)^{\varepsilon}$$ be the $\varepsilon$-invariant sublattice of the cohomology lattice $H^2(\tilde{S},\Z)$ and let $N:= M^\perp$ be its orthogonal complement. The direct orthogonal sum $M \oplus N$ is a finite index sublattice of the lattice $H^2(\tilde{S},\Z)$. Moreover, we know by [@namikawa Proposition (2.3)] that $M$ coincides with the pullback of $H^2(S,\Z)$ by $\pi$, hence we have the isomorphisms $$\label{eq-enriques-isomorphisms}
M \simeq \mathrm{Num}(S)(2) \simeq U(2)\oplus E_8(2) \text{ and } N\simeq U\oplus U(2)\oplus E_8(2),$$ where $U$ denotes the unimodular hyperbolic plane and $E_8$ is the unique even unimodular negative-definite lattice of rank 8. Moreover, for a lattice $L$ and $n\in \Q$, $L(n)$ denotes the lattice whose underlying abelian group is the same as $L$ and the bilinear form is multiplied by $n$. Basic facts concerning integral symmetric bilinear forms can be found in [@nikulin-sym].

Since the lift $\tilde{f}$ commutes with the involution $\varepsilon$, the map it induces on the cohomology lattice preserves sublattices $M$ and $N$. We put $${f_M}:=  \tilde f^* \rvert_M \mbox{ and } {f_N}:= \tilde f^* \rvert_N  \, .$$ Obviously, ${f_N}$ induces an isometry of the quadratic space $N \otimes \R$ of signature $(2,10)$ that preserves the original lattice $N =: N_{\Z} \subset N \otimes \R$ and the Hodge structure of $N$. Since the latter is exactly given by an oriented positive 2-plane in $N \otimes \R$, we have $${f_N}\in O(N_{\Z})\cap (O(2)\times O(10)).$$ Since the right-hand side is a discrete subgroup in a compact group, we obtain the following well-known fact.

\[Nfin\] The map ${f_N}$ is of finite order.

By Lemma \[Nfin\], the characteristic polynomial of ${f_N}$ is a product $$\label{eq-charpolfac}
{p}_N(x)=\det (xI-{f_N}) = \prod_{i=1}^k \Phi_{n_i}(x)$$ of cyclotomic polynomials $\Phi_{n_i}(x)$ for a collection of positive integers $\{n_i :i=1, \ldots, k\}$ with $\sum_{i=1}^k {\varphi}(n_i) = 12$, where ${\varphi}(\cdot)$ stands for the Euler totient function. Obviously, the order of ${f_N}$ is just the least common multiple $$\label{eq-ord-tdfn}
\ord({f_N}) = \operatorname{lcm}\{n_i, i = 1, \ldots,k\}.$$

The integers $m$ for which ${\varphi}(m)\leq 12$ are as follows.

   \[table-pphi\] ${\varphi}(m)$  $m$
  ------------------------------- ---------------------
               $12$               $13,21,26,28,36,42$
               $10$               $11,22$
               $ 8$               $15,16,20,24,30$
               $ 6$               $7,9,14,18$
               $ 4$               $5,8,10,12$
               $ 2$               $3,4,6$
               $ 1$               $1,2$

The proof of Thm \[th1\] will be based on the following lemmas.

\[lem1-new\] Let ${p}_N(x)$ be the characteristic polynomial of the map ${f_N}$.

[(a)]{} The reduction $({p}_N(x) \bmod 2)$ is divisible by $(x^2+1)=(x+1)^2$.

[(b)]{} ${p}_N(1) {p}_N(-1)$ is either zero or a square in $\mathbb{Q}^{*}$.

\(a) Let us consider the action of ${f_N}$ on the reduction $N\otimes \mathbb{F}_2 \cong (1/2)N/N$. Obviously, the reduction contains the 10-dimensional ${f_N}$-invariant subspace $$\label{eq-10dim}
N^*/N \subset(1/2)N/N,$$ where $N^*/N$ is the discriminant group of $N$. Thus the reduction $({p}_N(x) \bmod 2)$ is divisible by a degree two polynomial.

In fact, there exists an $11$-dimensional canonical subspace $N^{*,+}$ of the reduction $(1/2)N/N$ containing $N^*/N$. We discuss as follows. The residue group $(1/2)N/N^*$ consists of four residue classes modulo $N^*$ and $N^*$ has the property that for all $y\in N^*$, $(y,y)\in \Z$ (namely $\delta (N)=0$ in Nikulin’s notation.) Hence, the induced quadratic form $(1/2)N/N^*\rightarrow \Q/\Z$ is well-defined. Among the four residue classes, there exists a unique nonzero element whose form value is $1/2$, which corresponds to $N^{*,+}$. (The idea of this proof parallels [@allcock]). Thus we obtain (a).

\(b) Claim follows from [@BF Proposition 5.1].

\[lem-yuya\] The order $\mathrm{ord}(f_N)$ cannot be one of the integers $11, 22, 35, 70$.

We put $p_M(x) = \sum_{i=0}^{10} a_i x^i$ (resp. $p_{N^*/N}(x)$) to denote the characteristic polynomial of ${f_M}$ on $M$ (resp. of the map induced by ${f_N}$ on the discriminant group $N^*/N$). Since $M \oplus N$ is a finite index sublattice of the unimodular lattice $H^2(\tilde{S}, \Z)$ we have a canonical isomorphism of the discriminant groups $$M^*/M \cong N^*/N \, ,$$ which is $({f_M},{f_N})$-equivariant. Moreover, from $M^*/M = \frac{1}{2}M/M$ and the inclusion we infer $$\label{eq-pnstarn}
p_{N^*/N}(x) = (p_M(x)  \bmod 2) \quad \mbox{and} \quad p_{N^*/N}(x) {\mid}(p_N(x) \bmod 2) .$$

Assume $\mathrm{ord}(f_N) \in \{35, 70\}$. Then, by and the table on p.  we have $$\label{eq-pnx}
p_N(x) = \Phi_{5 l_1}(x) \Phi_{7 l_2}(x) \Phi_{l_3}(x) \Phi_{l_4}(x) \quad \mbox{ with }  l_1, l_2, l_3, l_4  \in \{1,2\} \, .$$ Thus implies that $$(p_M(x)  \bmod 2) = p_{N^*/N}(x) = F_5(x) F_7(x) = x^{10} + x^8 + x^6 + x^5 + x^4 + x^2 + 1 ,$$ so the coefficient $a_5$ and the sums $a_0 + a_2 + a_4$, $a_6 + a_8 + a_{10}$ are odd integers. In particular, the polynomial $p_M(x)$ is self-reciprocal (see e.g. [@cantat]), so the product $p_M(1)p_M(-1)$ can be expressed as $$\begin{aligned}
 \bigl(\sum_{i:even} a_i\bigr)^2 - \bigl(\sum_{i:odd}a_i\bigr)^2 &=& (2(a_0 + a_2 + a_4))^2 - (2(a_1 + a_3) + a_5)^2   \\ 
                 &=&   (2 \bmod 4)^2 - (1 \bmod 2)^2  \end{aligned}$$ Thus $p_M(1)p_M(-1)  \equiv 3 \bmod 8$, which contradicts [@BF Proposition 5.1].

If $\mathrm{ord}(f_N) \in \{11, 22\}$, then the factorization of $p_N(x)$ is as follows $$\label{eq-pnx11}
p_N(x) = \Phi_{11 l_1}(x) \Phi_{l_2}(x) \Phi_{l_3}(x) \quad \mbox{ with }  l_1, l_2, l_3  \in \{1,2\} \, .$$ Thus we have $(p_M(x)  \bmod 2) = F_{11}(x) $ and, as in the previous case we obtain $p_M(1)p_M(-1)  \equiv 3 \bmod 8$.

The proof of Lemma \[lem-yuya\] shows that the characteristic polynomial $p_M(x) = \sum_{i=0}^{10} a_i x^i$ of ${f_M}$ cannot be a polynomial such that the coefficient $a_5$ and the sums $a_0 + a_2 + a_4 = a_6 + a_8 + a_{10}$ are odd integers, i.e. the modulo 2 reduction $(p_M(x)  \bmod 2)$ cannot be one of the polynomials $$F_5(x) F_7(x), F_{11}(x), 
F_3(x)^5, F_3(x)^2 F_9(x), F_3(x) F_5(x)^2, F_3(x) F_{15}(x) \, .$$

The next lemma rules out the first two lines of the table on p. . It is also of use in the next section.

\[lem12\] If $\Phi_{m}(x)$ comes up in the factorization , then $${\varphi}(m) < 10 \, .$$

If ${\varphi}(m)=12=\operatorname{rank}N$, then the characteristic polynomial ${p}_N(x)$ equals the cyclotomic polynomial $\Phi_m(x)$. The decomposition of $\Phi_m(x) \bmod 2$ into irreducible factors is given in the table below:

  $m$     irreducible decomposition of $\Phi_m \bmod 2$
  ------- -----------------------------------------------
  42,21   $(x^6+x^4+x^2+x+1)(x^6+x^5+x^4+x^2+1)$
  36      $(x^6 + x^3 + 1)^2$
  28      $(x^3 + x + 1)^2 (x^3 + x^2 + 1)^2$
  26,13   $x^{12} + x^{11} + \cdots + x + 1$

Thus ${\varphi}(m) < 12$ by Lemma \[lem1-new\].a.

Suppose that ${\varphi}(m)=10$, namely $m=11,22$. Then $\mathrm{ord}(f_N)$ is a multiple of 11, and taking some power we get a contradiction to Lemma \[lem-yuya\].

After these preparations we can give the proof of Thm \[th1\].

Let $p(x) = \prod_{i=1}^k \Phi_{n_i}(x)$ be a product of cyclotomic polynomials for some $n_1$, $\ldots$ $n_k \in \mathbb{N}$ such that $\sum_{i=1}^k {\varphi}(n_i) = 12$ and ${\varphi}(n_i) < 10$ for $i=1, \ldots, k$. If $p(x)$ is the characteristic polynomial of the map ${f_N}$ induced by an automorphism of an Enriques surface, then it satisfies the conditions (a),(b) of Lemma \[lem1-new\] and the order of ${f_N}$ is given by . An enumeration of all cases (by hand or by a help of a computer) and Lemma \[lem-yuya\] show that $n =  \operatorname{lcm}\{n_i, i = 1, \ldots,k\}$ is one of the integers that appear in Theorem \[th1\].

\[rem-finite-ohashi\] To illustrate Theorem \[th1\], we shall give a classification of the case when the order of $f\in \mathrm{Aut}(S)$ is finite. Such automorphisms were classified in [@MO1; @ohashi-birep]. The following table gives the complete classification of the pair $(\mathrm{ord} (f),\mathrm{ord} (f_N))$.

    $\mathrm{ord} (f)$     1     2     3      4      5      6     8
  ---------------------- ----- ----- ----- ------- ------ ----- -----
   $\mathrm{ord} (f_N)$   1,2   1,2   3,6   1,2,4   5,10   3,6   4,8

We list here nontrivial examples exhibiting the pair $(\mathrm{ord} (f),\mathrm{ord} (f_N))$ and leave the proofs to the reader. The pair $(2,1)$ is supplied by No. 18 in [@IO]. When $\mathrm{ord} (f)=3$, $5$ or $6$, $f$ is semi-symplectic by [@MO1 Proposition 4.5] (i.e. $f$ acts trivially on the space $H^0(S, \mathcal{O}(2K_S))$) and we can use the symplectic lift to compute eigenvalues. Examples 1.1 and 1.2 of [@ohashi-birep] give the pairs $(4,4)$ and $(8,8)$. Finally, Example 1.3 of [@ohashi-birep] provides the remaining possibilities for $\mathrm{ord} (f)=4$ or $8$.

As Example \[example-existence-3factors\] shows, the order $\mathrm{ord} (f_N)$ is no longer bounded by $10$ when the order of $f \in \mathrm{Aut}(S)$ is infinite (see ). However, we have very few examples: the question of determining the exact list of possible $\ord({f_N})$ remains open.

Dynamical degrees {#sect-dynamical-degrees}
=================

We maintain the notation of the previous section. For the convenience of the reader we recall the definition of the dynamical degree.

Let $f \in \mathrm{Aut}(S)$. The [**dynamical degree $\lambda(f)$ of**]{} $f$ is defined as the spectral radius of the map $f^*:\mathrm{Num}(S) \rightarrow  \mathrm{Num}(S)$. One can show that the map $f^*$ has either none or exactly two eigenvalues away from the unit circle in $\C$. If such two eigenvalues come up, they are real and reciprocal. Thus either $\lambda(f) = 1$ or it is the largest real eigenvalue of the map $f^*$ (for a precise discussion of the above notion and its properties see [@cantat $\S$.2.2.2], [@mcmullen11], [@mcmullen16] and references therein).

After these preparations we are in position to give the proof of Thm \[mod2\] (c.f. [@third proof of Lemma 4.1]).

We put $p_M$ (resp. $p_N$, resp. $p_f$) to denote the characteristic polynomial of ${f_M}$ on $M$ (resp. ${f_N}$ on $N$, resp. $f^*$ on $\mathrm{Num}(S)$). We assume that $\lambda(f) \neq 1$ and denote the minimal polynomial of $\lambda(f)$ by $s_\lambda$.

By the first isomorphism in , the action of $f^*$ on the discriminant group of the lattice $\mathrm{Num}(S)(2)$ coincides with the action of ${f_M}$ on the discriminant group $M^*/M$. From $M^*/M = \frac{1}{2}M/M$, we obtain the equality of modulo 2 reductions: $$p_M \equiv p_f \bmod 2 .$$ Moreover yields: $$(p_M  \bmod 2) {\mid}(p_N \bmod 2) .$$

Let $h$ be an irreducible factor of the reduction $(s_\lambda \bmod 2)$. We have just shown $h$ appears also in the factorization of $(p_N \bmod 2)$. Then, from and Lemma \[lem12\], $h$ divides $(\Phi_m \bmod 2)$ for certain $m$ such that ${\varphi}(m) \leq 8$. Since $(\Phi_{2^e m} \bmod 2)$ is a power of the reduction $(\Phi_m \bmod 2)$, we may assume $m$ to be odd. Hence, by the table on p. , we have $m = 1,3,5,7,9,15$. Thus either $h = F_m$, with $m=1,3,5,9$ or $h$ appears in the factorization of $F_m$, where $m=7,15$. But, for $m=7,15$ we have $F_m = F_{m,1} \cdot F_{m,2}$ where $$\begin{aligned}
F_{7,1} :=  (x^3 + x + 1),                           
& \quad  F_{7,2}  := (x^3 + x^2 + 1),   \\
F_{15,1} :=   (x^4 + x + 1),            
& \quad F_{15,2}  := (x^4 + x^3 + 1).\end{aligned}$$ Being a Salem polynomial, $s_\lambda$ is self-reciprocal, and so is its modulo $2$ reduction. Since the polynomials $F_{7,1}$ and $F_{7,2}$ are not self-reciprocal, their multiplicities in $(s_\lambda \bmod 2)$ should coincide with those of their reciprocal counterparts. The same holds for $F_{15,1}$ and $F_{15,2}$. This completes the proof.

It is natural to ask which of the six factors given in Thm \[mod2\] do appear in modulo 2 reductions of minimal polynomials of dynamical degrees of automorphisms of Enriques surfaces. We do not know whether the polynomials $F_{7}(x)$, $F_{9}(x)$, $F_{15}(x)$ are realized by automorphisms of Enriques surfaces. To answer the question whether $F_m$ where $m=1,3,5$ come up in $(s_{\lambda} \bmod 2)$, we analyze some automorphisms constructed in [@dolgachev16].

\[example-existence-3factors\] (a) By [@dolgachev16 Sect. 4.5, Table 2] there exists an Enriques surface $S$ of Hesse type and $f \in \mathrm{Aut}(S)$ such that the characteristic polynomial $p_{f^*}(x)$ of $f^* \in \mathrm{Aut}(\mathrm{Num}(S))$ equals[^2] $$\label{eq-f5f3}
p_{f^*}(x) = x^{10} - 6x^9 - 7x^8 - 9x^7 - 6x^6 - 10x^5 - 6x^4 - 9x^3 - 7x^2 - 6x +1 .$$ One can check $p_{f^*}(x) \in \Z[x]$ is irreducible and we have $$(p_{f^*}(x)  \bmod 2) = F_5(x) \cdot F_3(x) \cdot F_1^4(x).$$ In particular, combined with the proof of Thm \[mod2\] yields that $$\label{eq-order15}
15 {\mid}\ord({f_N}).$$

\(b) According to [@dolgachev16 Sect. 3.2, Case $m = 4$] there exists an Enriques surface $S$ and $f \in \mathrm{Aut}(S)$ such that $p_{f^*}(x)$ is divisible by the following polynomial $$\label{eq-dolgtocheck}
x^8 - 165x^7 + 223x^6 - 59x^5 - 133x^4 - 59x^3 + 223x^2 - 165x + 1 \, .$$ Since the modulo 2 reduction of the above polynomial factors as the product $(F_3(x) \cdot F_9(x))$, this would imply that $F_9(x)$ can also appear in the reduction of the minimal polynomial $s_{\lambda}$. Unfortunately, there is a misprint in [@dolgachev16 Sect. 3.2, Case $m = 4$]: the term $x^4$ comes with the coefficient $(-144)$ (instead of $(-133)$ as in ). Thus the question whether the factor $F_9(x)$ is possible or not remains open.

The example below shows that even the direct potential refinement of Thm \[mod2\] (i.e. ruling out all/some of the factors $F_7$, $F_9$, $F_{15}$ in the modulo 2 reduction of the characteristic polynomial) cannot lead to a necessary and sufficient condition for a Salem number to be the dynamical degree of an automorphism of an Enriques surface.

\[example-strange\] Consider the Salem number $\lambda = 1.64558...$ given by the polynomial $$\label{eq-mini-salem}
s_{\lambda} := x^{10}-x^9-x^8-x^2-x+1 \, .$$ As one can easily check, we have $$(s_{\lambda}  \bmod 2) =  F_3(x) \cdot F_1^8(x).$$ Thus Thm \[mod2\] does not rule out the above number as the dynamical degree of an automorphism of an Enriques surface. But we can apply [@Gross--McMullen Theorem 6.1]. Suppose that the Salem number given by is the dynamical degree of an automorphism of an Enriques surface $S$. The lattice $\mbox{Num}(S)$ is of rank 10, so $s_{\lambda}$ is the full characteristic polynomial of the induced automorphism on $\mbox{Num}(S)$. Also $\mathrm{Num}(S)$ is even unimodular, so by [@Gross--McMullen Theorem 6.1] both $|s_{\lambda}(\pm 1)|$ must be squares, which is not the case. This contradiction shows that cannot be the minimal polynomial of the dynamical degree of an automorphism of an Enriques surface.

Presently, the smallest known non-trivial dynamical degree is the one constructed by Dolgachev:

\[Dolgachevexample\] By [@dolgachev16 Sect. 4.5, Table 2] there exists an Enriques surface $S$ of Hesse type and $f \in \mathrm{Aut}(S)$ such that the characteristic polynomial $p_{f^*}(x)$ of $f^* \in \mathrm{Aut}(\mathrm{Num}(S))$ has the Salem polynomial $$\label{eq-mp-lambdaD}
x^4-x^3-2x^2-x+1$$ as a factor. The largest real root of the above polynomial is $$\lambda_D := 2.08101... \, .$$ This gives the smallest known value of non-trivial dynamical degree of an automorphism of an Enriques surface.

\[MO15family\] As an another example, let us consider the Enriques surface $S$ in the family of [@MO15]. In that paper, it is proved that $\mathrm{Aut}(S)\simeq C_2^{*4}\rtimes \mathfrak{S}_4$. We here prove that every element in this group has dynamical degree at least $\lambda_D$.

In fact, [@MO15] exhibits 10 smooth rational curves $E_i,\ E_{ij}\ (i,j=1,\dots,4,\ i\neq j)$ on $S$ which form a rational basis of $\mathrm{Num}(S)$. We denote by $L$ the sublattice generated by them. The symmetric group $\mathfrak{S}_4$ acts on the indices of generators, while the generators $s_i$ of the four cyclic groups $C_2$ act by reflections in some divisors $G_i\in L$ of self-intersection $-2$. In particular, $\mathrm{Aut}(S)$ preserves $L$. We use the relations $(G_i, E_{kl})=0$ (for all $i,k,l$) and $(G_i, l)\in 2\mathbb{Z}$ for all $l\in L$.

Now from $(G_i, E_{kl})=0$, the six-dimensional subspace $V$ generated by $E_{ij}$ is stable under $\mathrm{Aut}(S)$. Since it is negative-definite, we see that any Salem number on $S$ has degree at most four, the dimension of the complement to $V$. Thus for any element in $\mathrm{Aut}(S)$, the characteristic polynomial $F$ decomposes into degree four (on $L_{\mathbb{C}}/V_{\mathbb{C}}$) and degree six (on $V_{\mathbb{C}}$). Moreover, since $G_i$ intersects evenly with all $l\in L$, it acts trivially on $L/2L$. Hence the degree four part of $(F \bmod 2)$ decomposes either into linear factors or has only one non-linear factor $x^2+x+1$, which arises when the residue class has order 3 in $\mathfrak{S}_4$. Looking through the list in the Appendix, we get the assertion. (This is something unfortunate, although.)

We use Theorem \[mod2\] on low-degree Salem numbers that do not exceed $\lambda_D$.

\[rem-possibilities\] One can check that there are exactly 133 Salem numbers of degree $\leq 10$ up to Dolgachev’s record $\lambda_D$. (The list of small Salem numbers can be found in [@mossinghoff].) Oguiso’s criterion [@third Lemma 4.1] shows that 28 among them cannot be dynamical degrees of automorphisms of Enriques surfaces. To go further, combining Theorem \[mod2\] and [@Gross--McMullen] we can prove that 65 more cannot be realized by automorphisms of Enriques surfaces. For the convenience of readers we list the 133 Salem numbers in question, their minimal polynomials and their modulo 2 reductions in $\S.$\[sect-appendix\] Appendix. Those numbers excluded by modulo 2 reductions are marked “impossible”. Non-existence that results from [@Gross--McMullen Theorem 6.1] (see Example \[example-strange\]) is marked “impossible (\*)”. (In some cases both apply.)

In conclusion, [*there remain $39$ Salem numbers as candidates for the minimal dynamical degree of automorphisms of Enriques surfaces.*]{}

On the arXiv[^3] we attach two lists, in plain text format, of the coefficients of the minimal polynomials of (1) these $39$ candidates, and (2) all $133$ Salem numbers of degree $\leq 10$ up to Dolgachev’s record $\lambda_D$.

Finally we can give the proof of Corollary \[th2\].

There are only finitely many Salem numbers less than $\lambda = 1.35098...$ and of degree $\leq 10$ (there are none of degree $\leq 6$, one of degree $8$, and six of degree $10$). By a straightforward calculation each of those numbers violates the condition of Thm \[mod2\].

For explicit factorization see the first seven entries of the table in $\S.$\[sect-appendix\] Appendix.

[**Acknowledgement:**]{} This project was completed during the workshop “New Trends in Arithmetic and Geometry of Algebraic Surfaces” in BIRS (Banff, Canada). The authors would like to thank the organizers and the staff of BIRS for very good working conditions. We thank the referee for inspiring comments and calling our attention to the paper [@BF].

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Appendix: List of low-degree Salem numbers up to $\lambda_D$ {#sect-appendix}
============================================================

Below we list all Salem numbers of degree $\leq 10$ up to the dynamical degree $\lambda_D$ of Dolgachev’s example and show the minimal polynomial $s_\lambda$ and its mod 2 factorization. Some of them are impossible by Thm \[mod2\] and marked “impossible”. Others marked “impossible (\*)” are those excluded by [@Gross--McMullen Theorem 6.1].

As noted in Remark \[rem-possibilities\], the list of the coefficients of the minimal polynomials are available in plain text format on the arXiv.

  ---- ----- ------------ ------------------------------------------------------------------ -------------------
    \#   deg value        minimal polynomial $s_\lambda$                                     
                          factorization of $s_\lambda \bmod 2$                               conclusion
          10 1.17628...   $x^{10}+x^9-x^7-x^6-x^5-x^4-x^3+x+1$                               
    \*                    $(x^5 + x^3 + x^2 + x + 1)  (x^5 + x^4 + x^3 + x^2 + 1)$           impossible
          10 1.21639...   $x^{10}-x^6-x^5-x^4+1$                                             
    \*                    $(x^5 + x^4 + x^2 + x + 1)  (x^5 + x^4 + x^3 + x + 1)$             impossible
          10 1.23039...   $x^{10}-x^7-x^5-x^3+1$                                             
    \*                    $x^{10} + x^7 + x^5 + x^3 + 1$                                     impossible
          10 1.26123...   $x^{10}-x^8-x^5-x^2+1$                                             
    \*                    $(x^2 + x + 1)  (x^8 + x^7 + x^6 + x^4 + x^2 + x + 1)$             impossible
           8 1.28063...   $x^8-x^5-x^4-x^3+1$                                                
    \*                    $x^8 + x^5 + x^4 + x^3 + 1$                                        impossible
          10 1.29348...   $x^{10}-x^8-x^7+x^5-x^3-x^2+1$                                     
    \*                    $(x^5 + x^2 + 1)  (x^5 + x^3 + 1)$                                 impossible
          10 1.33731...   $x^{10}-x^9-x^5-x+1$                                               
    \*                    $x^{10} + x^9 + x^5 + x + 1$                                       impossible
     1    10 1.35098...   $x^{10}-x^9-x^6+x^5-x^4-x+1$                                       
    \*                    $(x^2 + x + 1)^2  (x^3 + x + 1)  (x^3 + x^2 + 1)$                  
           8 1.35999...   $x^8-x^7+x^6-2x^5+x^4-2x^3+x^2-x+1$                                
    \*                    $x^8 + x^7 + x^6 + x^4 + x^2 + x + 1$                              impossible
          10 1.38363...   $x^{10}-x^9-x^7+x^6-x^5+x^4-x^3-x+1$                               
    \*                    $(x^5 + x^3 + x^2 + x + 1)  (x^5 + x^4 + x^3 + x^2 + 1)$           impossible
     2     6 1.40126...   $x^6-x^4-x^3-x^2+1$                                                
    \*                    $(x^2 + x + 1)  (x^4 + x^3 + x^2 + x + 1)$                         
     3     8 1.42500...   $x^8-x^7-x^5+x^4-x^3-x+1$                                          
    \*                    $(x^4 + x + 1)  (x^4 + x^3 + 1)$                                   
          10 1.43100...   $x^{10}-x^9-x^8+x^7-x^5+x^3-x^2-x+1$                               
    \*                    $(x^2 + x + 1)  (x^8 + x^5 + x^4 + x^3 + 1)$                       impossible
          10 1.44842...   $x^{10}-2x^9+2x^8-2x^7+x^6-x^5+x^4-2x^3+2x^2-2x+1$                 
    \*                    $(x^5 + x^4 + x^2 + x + 1)  (x^5 + x^4 + x^3 + x + 1)$             impossible
     4     8 1.45798...   $x^8-x^6-x^5-x^3-x^2+1$                                            
    \*                    $(x + 1)^2  (x^6 + x^3 + 1)$                                       
          10 1.47235...   $x^{10}-x^9-x^6-x^4-x+1$                                           
    \*                    $(x + 1)^6  (x^4 + x^3 + x^2 + x + 1)$                             impossible (\*)
          10 1.47960...   $x^{10}-2x^8-2x^7+x^6+3x^5+x^4-2x^3-2x^2+1$                        
    \*                    $(x^5 + x^4 + x^2 + x + 1)  (x^5 + x^4 + x^3 + x + 1)$             impossible
     5     6 1.50613...   $x^6-x^5-x^3-x+1$                                                  
    \*                    $(x^2 + x + 1)^3$                                                  
          10 1.51386...   $x^{10}-x^7-2x^6-x^5-2x^4-x^3+1$                                   
    \*                    $x^{10} + x^7 + x^5 + x^3 + 1$                                     impossible
           8 1.52306...   $x^8-x^7-x^6+x^4-x^2-x+1$                                          
    \*                    $x^8 + x^7 + x^6 + x^4 + x^2 + x + 1$                              impossible
     6    10 1.53292...   $x^{10}-x^9-x^8+x^5-x^2-x+1$                                       
    \*                    $(x^4 + x^3 + x^2 + x + 1)  (x^6 + x^3 + 1)$                       
           8 1.54719...   $x^8-2x^7+2x^6-3x^5+3x^4-3x^3+2x^2-2x+1$                           
    \*                    $x^8 + x^5 + x^4 + x^3 + 1$                                        impossible
     7     6 1.55603...   $x^6-x^5-x^4+x^3-x^2-x+1$                                          
    \*                    $(x^3 + x + 1)  (x^3 + x^2 + 1)$                                   
     8     6 1.58234...   $x^6-x^4-2x^3-x^2+1$                                               
    \*                    $(x + 1)^6$                                                        
          10 1.59070...   $x^{10}-2x^9+x^8-2x^6+3x^5-2x^4+x^2-2x+1$                          
    \*                    $(x^2 + x + 1)  (x^8 + x^7 + x^6 + x^4 + x^2 + x + 1)$             impossible
          10 1.59700...   $x^{10}-x^8-x^7-x^6-x^5-x^4-x^3-x^2+1$                             
    \*                    $(x^2 + x + 1)^2  (x^6 + x^3 + 1)$                                 impossible (\*)
          10 1.59866...   $x^{10}-2x^9+x^8-x^7+2x^6-3x^5+2x^4-x^3+x^2-2x+1$                  
    \*                    $(x^5 + x^2 + 1)  (x^5 + x^3 + 1)$                                 impossible
     9     8 1.60544...   $x^8-2x^7+x^6-x^4+x^2-2x+1$                                        
    \*                    $(x^4 + x^3 + x^2 + x + 1)^2$                                      
          10 1.62501...   $x^{10}-x^9-x^8-x^7+x^6+x^5+x^4-x^3-x^2-x+1$                       
    \*                    $x^{10} + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1$   impossible (\*)
          10 1.62754...   $x^{10}-2x^9+2x^7-x^6-x^5-x^4+2x^3-2x+1$                           
    \*                    $(x^5 + x^4 + x^2 + x + 1)  (x^5 + x^4 + x^3 + x + 1)$             impossible
    10     6 1.63557...   $x^6-2x^5+2x^4-3x^3+2x^2-2x+1$                                     
    \*                    $x^6 + x^3 + 1$                                                    
           8 1.64003...   $x^8-2x^6-x^5+x^4-x^3-2x^2+1$                                      
    \*                    $x^8 + x^5 + x^4 + x^3 + 1$                                        impossible
          10 1.64558...   $x^{10}-x^9-x^8-x^2-x+1$                                           
    \*                    $(x + 1)^8  (x^2 + x + 1)$                                         impossible (\*)
          10 1.65740...   $x^{10}+x^9-2x^7-4x^6-5x^5-4x^4-2x^3+x+1$                          
    \*                    $x^{10} + x^9 + x^5 + x + 1$                                       impossible
    11     8 1.66104...   $x^8-2x^7+x^6-x^5+x^4-x^3+x^2-2x+1$                                
    \*                    $(x^2 + x + 1)  (x^3 + x + 1)  (x^3 + x^2 + 1)$                    
          10 1.66929...   $x^{10}-x^9-x^7-2x^5-x^3-x+1$                                      
    \*                    $(x + 1)^2  (x^8 + x^7 + x^6 + x^4 + x^2 + x + 1)$                 impossible
          10 1.67310...   $x^{10}-2x^9+x^8-x^7+x^5-x^3+x^2-2x+1$                             
    \*                    $(x^5 + x^2 + 1)  (x^5 + x^3 + 1)$                                 impossible
    12     8 1.68491...   $x^8-x^7-x^6-x^2-x+1$                                              
    \*                    $(x + 1)^2  (x^2 + x + 1)^3$                                       
          10 1.69017...   $x^{10}-x^9-2x^8+x^7+x^6-x^5+x^4+x^3-2x^2-x+1$                     
    \*                    $(x^5 + x^3 + x^2 + x + 1)  (x^5 + x^4 + x^3 + x^2 + 1)$           impossible
    13     8 1.69350...   $x^8-x^7-x^5-x^4-x^3-x+1$                                          
    \*                    $(x^4 + x + 1)  (x^4 + x^3 + 1)$                                   
          10 1.71336...   $x^{10}-2x^9+x^8-2x^6+2x^5-2x^4+x^2-2x+1$                          
    \*                    $(x + 1)^{10}$                                                     impossible (\*)
    14     4 1.72208...   $x^4-x^3-x^2-x+1$                                                  
    \*                    $x^4 + x^3 + x^2 + x + 1$                                          
          10 1.73694...   $x^{10}-x^9-x^7-x^6-x^5-x^4-x^3-x+1$                               
    \*                    $(x^5 + x^3 + x^2 + x + 1)  (x^5 + x^4 + x^3 + x^2 + 1)$           impossible
          10 1.74492...   $x^{10}-2x^9+2x^8-3x^7+2x^6-3x^5+2x^4-3x^3+2x^2-2x+1$              
    \*                    $x^{10} + x^7 + x^5 + x^3 + 1$                                     impossible
          10 1.74601...   $x^{10}-x^9-x^8-x^7+2x^5-x^3-x^2-x+1$                              
    \*                    $(x + 1)^4  (x^3 + x + 1)  (x^3 + x^2 + 1)$                        impossible (\*)
          10 1.75173...   $x^{10}-2x^9+x^8-x^7+x^6-2x^5+x^4-x^3+x^2-2x+1$                    
    \*                    $(x + 1)^2  (x^8 + x^5 + x^4 + x^3 + 1)$                           impossible
          10 1.75309...   $x^{10}-x^8-x^7-2x^6-3x^5-2x^4-x^3-x^2+1$                          
    \*                    $(x^5 + x^2 + 1)  (x^5 + x^3 + 1)$                                 impossible
          10 1.76015...   $x^{10}-2x^9+x^8-2x^7+2x^6-x^5+2x^4-2x^3+x^2-2x+1$                 
    \*                    $(x^2 + x + 1)  (x^8 + x^7 + x^6 + x^4 + x^2 + x + 1)$             impossible
          10 1.76400...   $x^{10}-2x^9+x^7-x^5+x^3-2x+1$                                     
    \*                    $x^{10} + x^7 + x^5 + x^3 + 1$                                     impossible
          10 1.76690...   $x^{10}-2x^8-2x^7+x^5-2x^3-2x^2+1$                                 
    \*                    $(x^2 + x + 1)  (x^4 + x + 1)  (x^4 + x^3 + 1)$                    impossible (\*)
          10 1.77056...   $x^{10}-3x^9+4x^8-5x^7+5x^6-5x^5+5x^4-5x^3+4x^2-3x+1$              
    \*                    $(x^5 + x^3 + x^2 + x + 1)  (x^5 + x^4 + x^3 + x^2 + 1)$           impossible
    15     6 1.78164...   $x^6-x^5-x^4-x^2-x+1$                                              
    \*                    $(x + 1)^4  (x^2 + x + 1)$                                         
          10 1.78840...   $x^{10}-x^9-2x^7-x^5-2x^3-x+1$                                     
    \*                    $x^{10} + x^9 + x^5 + x + 1$                                       impossible
           8 1.79607...   $x^8-x^7-x^6-x^4-x^2-x+1$                                          
    \*                    $x^8 + x^7 + x^6 + x^4 + x^2 + x + 1$                              impossible
          10 1.79978...   $x^{10}-3x^8-3x^7+2x^6+5x^5+2x^4-3x^3-3x^2+1$                      
    \*                    $(x^5 + x^2 + 1)  (x^5 + x^3 + 1)$                                 impossible
    16     8 1.80017...   $x^8-3x^7+4x^6-5x^5+5x^4-5x^3+4x^2-3x+1$                           
    \*                    $(x^4 + x + 1)  (x^4 + x^3 + 1)$                                   
    17    10 1.80501...   $x^{10}-2x^9+x^7-x^6+x^5-x^4+x^3-2x+1$                             
    \*                    $(x^2 + x + 1)^3  (x^4 + x^3 + x^2 + x + 1)$                       
    18     8 1.80978...   $x^8-x^7-2x^5-2x^3-x+1$                                            
    \*                    $(x + 1)^2  (x^3 + x + 1)  (x^3 + x^2 + 1)$                        
           8 1.81161...   $x^8-2x^7+x^5-x^4+x^3-2x+1$                                        
    \*                    $x^8 + x^5 + x^4 + x^3 + 1$                                        impossible
          10 1.82383...   $x^{10}-x^8-2x^7-2x^6-2x^5-2x^4-2x^3-x^2+1$                        
    \*                    $(x + 1)^{10}$                                                     impossible (\*)
    19    10 1.82514...   $x^{10}-x^9-2x^8+x^6+x^5+x^4-2x^2-x+1$                             
    \*                    $(x^2 + x + 1)^2  (x^3 + x + 1)  (x^3 + x^2 + 1)$                  
    20     6 1.83107...   $x^6-2x^5+x^3-2x+1$                                                
    \*                    $x^6 + x^3 + 1$                                                    
    21     8 1.83488...   $x^8-x^6-2x^5-3x^4-2x^3-x^2+1$                                     
    \*                    $(x^4 + x^3 + x^2 + x + 1)^2$                                      
          10 1.84835...   $x^{10}-x^9-x^8-x^6-x^5-x^4-x^2-x+1$                               
    \*                    $(x^2 + x + 1)^5$                                                  impossible (\*)
           8 1.84959...   $x^8+x^7-x^6-4x^5-5x^4-4x^3-x^2+x+1$                               
    \*                    $x^8 + x^7 + x^6 + x^4 + x^2 + x + 1$                              impossible
          10 1.85312...   $x^{10}-2x^9+x^8-2x^7+2x^6-2x^5+2x^4-2x^3+x^2-2x+1$                
    \*                    $(x + 1)^{10}$                                                     impossible (\*)
          10 1.85712...   $x^{10}-2x^9+x^8-x^7-x^5-x^3+x^2-2x+1$                             
    \*                    $(x^5 + x^2 + 1)  (x^5 + x^3 + 1)$                                 impossible
          10 1.86264...   $x^{10}-3x^9+3x^8-x^7-3x^6+5x^5-3x^4-x^3+3x^2-3x+1$                
    \*                    $x^{10} + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1$   impossible (\*)
    22     8 1.86406...   $x^8-x^7-2x^6+2x^4-2x^2-x+1$                                       
    \*                    $(x + 1)^2  (x^3 + x + 1)  (x^3 + x^2 + 1)$                        
          10 1.86876...   $x^{10}-x^9-2x^7-3x^5-2x^3-x+1$                                    
    \*                    $x^{10} + x^9 + x^5 + x + 1$                                       impossible
          10 1.87573...   $x^{10}-2x^9-x^8+3x^7-3x^5+3x^3-x^2-2x+1$                          
    \*                    $(x^5 + x^2 + 1)  (x^5 + x^3 + 1)$                                 impossible
    23     4 1.88320...   $x^4-2x^3+x^2-2x+1$                                                
    \*                    $(x^2 + x + 1)^2$                                                  
          10 1.88996...   $x^{10}-x^9-x^8-x^6-2x^5-x^4-x^2-x+1$                              
    \*                    $(x + 1)^2  (x^4 + x + 1)  (x^4 + x^3 + 1)$                        impossible (\*)
          10 1.89360...   $x^{10}-x^9-2x^8+x^6+x^4-2x^2-x+1$                                 
    \*                    $(x + 1)^6  (x^4 + x^3 + x^2 + x + 1)$                             impossible (\*)
          10 1.89663...   $x^{10}-2x^9+x^7-x^6-x^4+x^3-2x+1$                                 
    \*                    $(x + 1)^4  (x^6 + x^3 + 1)$                                       impossible (\*)
          10 1.89910...   $x^{10}-2x^9+x^6-x^5+x^4-2x+1$                                     
    \*                    $(x^5 + x^4 + x^2 + x + 1)  (x^5 + x^4 + x^3 + x + 1)$             impossible
          10 1.90562...   $x^{10}-x^8-2x^7-3x^6-3x^5-3x^4-2x^3-x^2+1$                        
    \*                    $(x^3 + x + 1)  (x^3 + x^2 + 1)  (x^4 + x^3 + x^2 + x + 1)$        impossible (\*)
          10 1.90830...   $x^{10}-x^9-x^8-x^7-x^5-x^3-x^2-x+1$                               
    \*                    $(x^2 + x + 1)  (x^8 + x^5 + x^4 + x^3 + 1)$                       impossible
          10 1.91112...   $x^{10}-2x^8-2x^7-x^6-x^5-x^4-2x^3-2x^2+1$                         
    \*                    $(x^5 + x^4 + x^2 + x + 1)  (x^5 + x^4 + x^3 + x + 1)$             impossible
          10 1.91445...   $x^{10}-x^9-x^7-3x^6-x^5-3x^4-x^3-x+1$                             
    \*                    $(x^5 + x^3 + x^2 + x + 1)  (x^5 + x^4 + x^3 + x^2 + 1)$           impossible
    24     8 1.91649...   $x^8-x^7-x^6-x^5-x^3-x^2-x+1$                                      
    \*                    $(x + 1)^4  (x^4 + x^3 + x^2 + x + 1)$                             
           8 1.92062...   $x^8-3x^7+3x^6-2x^5+x^4-2x^3+3x^2-3x+1$                            
    \*                    $x^8 + x^7 + x^6 + x^4 + x^2 + x + 1$                              impossible
          10 1.92606...   $x^{10}-x^9-x^8-x^7-x^6+x^5-x^4-x^3-x^2-x+1$                       
    \*                    $x^{10} + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1$   impossible (\*)
    25     8 1.92678...   $x^8-2x^6-2x^5-x^4-2x^3-2x^2+1$                                    
    \*                    $(x^2 + x + 1)^4$                                                  
          10 1.92990...   $x^{10}-x^9-x^8-2x^7+x^6+x^4-2x^3-x^2-x+1$                         
    \*                    $(x + 1)^2  (x^4 + x + 1)  (x^4 + x^3 + 1)$                        impossible (\*)
          10 1.93231...   $x^{10}-2x^9+2x^8-4x^7+3x^6-5x^5+3x^4-4x^3+2x^2-2x+1$              
    \*                    $(x^5 + x^4 + x^2 + x + 1)  (x^5 + x^4 + x^3 + x + 1)$             impossible
          10 1.93295...   $x^{10}-3x^9+3x^8-2x^7+x^5-2x^3+3x^2-3x+1$                         
    \*                    $(x^4 + x^3 + x^2 + x + 1)  (x^6 + x^3 + 1)$                       impossible (\*)
          10 1.93637...   $x^{10}-2x^9+x^5-2x+1$                                             
    \*                    $(x^2 + x + 1)  (x^4 + x + 1)  (x^4 + x^3 + 1)$                    impossible (\*)
          10 1.94005...   $x^{10}-2x^9+x^8-x^7-x^6-x^4-x^3+x^2-2x+1$                         
    \*                    $(x + 1)^2  (x^8 + x^5 + x^4 + x^3 + 1)$                           impossible
    26     6 1.94685...   $x^6-x^5-x^4-x^3-x^2-x+1$                                          
    \*                    $(x^3 + x + 1)  (x^3 + x^2 + 1)$                                   
          10 1.94998...   $x^{10}-x^9-2x^8-x^7+x^6+3x^5+x^4-x^3-2x^2-x+1$                    
    \*                    $(x^5 + x^3 + x^2 + x + 1)  (x^5 + x^4 + x^3 + x^2 + 1)$           impossible
           8 1.95530...   $x^8-2x^7-x^5+3x^4-x^3-2x+1$                                       
    \*                    $x^8 + x^5 + x^4 + x^3 + 1$                                        impossible
    27     6 1.96355...   $x^6-2x^5-x^4+3x^3-x^2-2x+1$                                       
    \*                    $(x^2 + x + 1)  (x^4 + x^3 + x^2 + x + 1)$                         
          10 1.97209...   $x^{10}-2x^9-x^6+3x^5-x^4-2x+1$                                    
    \*                    $(x^5 + x^4 + x^2 + x + 1)  (x^5 + x^4 + x^3 + x + 1)$             impossible
    28     6 1.97481...   $x^6-2x^5+x^4-2x^3+x^2-2x+1$                                       
    \*                    $(x + 1)^6$                                                        
    29     6 1.98779...   $x^6-2x^4-3x^3-2x^2+1$                                             
    \*                    $x^6 + x^3 + 1$                                                    
    30     8 1.99400...   $x^8-2x^7+x^6-2x^5+x^4-2x^3+x^2-2x+1$                              
    \*                    $(x^4 + x^3 + x^2 + x + 1)^2$                                      
          10 1.99703...   $x^{10}-x^9-x^8-x^7-x^6-x^5-x^4-x^3-x^2-x+1$                       
    \*                    $x^{10} + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1$   impossible (\*)
    31    10 1.99852...   $x^{10}-2x^9+x^8-2x^7+x^6-2x^5+x^4-2x^3+x^2-2x+1$                  
    \*                    $(x + 1)^2  (x^2 + x + 1)^4$                                       
          10 2.00145...   $x^{10}-x^9-2x^8-x^7+x^6+2x^5+x^4-x^3-2x^2-x+1$                    
    \*                    $(x + 1)^4  (x^2 + x + 1)^3$                                       impossible (\*)
          10 2.00289...   $x^{10}-3x^9+3x^8-3x^7+3x^6-3x^5+3x^4-3x^3+3x^2-3x+1$              
    \*                    $x^{10} + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1$   impossible (\*)
          10 2.00573...   $x^{10}-2x^9-2x+1$                                                 
    \*                    $(x + 1)^2  (x^4 + x^3 + x^2 + x + 1)^2$                           impossible (\*)
          10 2.00624...   $x^{10}-x^8-3x^7-3x^6-4x^5-3x^4-3x^3-x^2+1$                        
    \*                    $(x + 1)^2  (x^8 + x^5 + x^4 + x^3 + 1)$                           impossible
          10 2.00947...   $x^{10}-2x^9+x^8-x^7-2x^6+x^5-2x^4-x^3+x^2-2x+1$                   
    \*                    $(x^5 + x^2 + 1)  (x^5 + x^3 + 1)$                                 impossible
    32     8 2.01128...   $x^8-3x^7+3x^6-3x^5+3x^4-3x^3+3x^2-3x+1$                           
    \*                    $(x^2 + x + 1)  (x^6 + x^3 + 1)$                                   
          10 2.01335...   $x^{10}-x^9-2x^7-2x^6-3x^5-2x^4-2x^3-x+1$                          
    \*                    $x^{10} + x^9 + x^5 + x + 1$                                       impossible
          10 2.01488...   $x^{10}-2x^8-2x^7-2x^6-3x^5-2x^4-2x^3-2x^2+1$                      
    \*                    $(x^2 + x + 1)  (x^4 + x + 1)  (x^4 + x^3 + 1)$                    impossible (\*)
          10 2.01600...   $x^{10}-3x^9+2x^8+x^7-3x^6+3x^5-3x^4+x^3+2x^2-3x+1$                
    \*                    $(x^5 + x^3 + x^2 + x + 1)  (x^5 + x^4 + x^3 + x^2 + 1)$           impossible
          10 2.01671...   $x^{10}-2x^9-x^8+2x^7+x^6-3x^5+x^4+2x^3-x^2-2x+1$                  
    \*                    $(x^3 + x + 1)  (x^3 + x^2 + 1)  (x^4 + x^3 + x^2 + x + 1)$        impossible (\*)
    33     8 2.02202...   $x^8-2x^7-2x+1$                                                    
    \*                    $(x + 1)^8$                                                        
          10 2.02344...   $x^{10}-2x^9-x^7+2x^6-x^5+2x^4-x^3-2x+1$                           
    \*                    $x^{10} + x^7 + x^5 + x^3 + 1$                                     impossible
          10 2.02739...   $x^{10}-x^9-x^8-x^7-x^6-2x^5-x^4-x^3-x^2-x+1$                      
    \*                    $(x + 1)^2  (x^2 + x + 1)^2  (x^4 + x^3 + x^2 + x + 1)$            impossible (\*)
    34     8 2.03064...   $x^8-x^7-3x^5-x^4-3x^3-x+1$                                        
    \*                    $(x^4 + x + 1)  (x^4 + x^3 + 1)$                                   
          10 2.03298...   $x^{10}-2x^9+x^6-3x^5+x^4-2x+1$                                    
    \*                    $(x^5 + x^4 + x^2 + x + 1)  (x^5 + x^4 + x^3 + x + 1)$             impossible
          10 2.03579...   $x^{10}-2x^9-x^6+2x^5-x^4-2x+1$                                    
    \*                    $(x + 1)^6  (x^2 + x + 1)^2$                                       impossible (\*)
          10 2.03890...   $x^{10}-x^9-2x^8-2x^7+2x^6+3x^5+2x^4-2x^3-2x^2-x+1$                
    \*                    $x^{10} + x^9 + x^5 + x + 1$                                       impossible
    35     6 2.04249...   $x^6-3x^5+3x^4-3x^3+3x^2-3x+1$                                     
    \*                    $(x^3 + x + 1)  (x^3 + x^2 + 1)$                                   
          10 2.04414...   $x^{10}-x^9-2x^8-x^7+x^6+x^5+x^4-x^3-2x^2-x+1$                     
    \*                    $(x^5 + x^3 + x^2 + x + 1)  (x^5 + x^4 + x^3 + x^2 + 1)$           impossible
          10 2.04776...   $x^{10}-3x^9+3x^8-3x^7+2x^6-x^5+2x^4-3x^3+3x^2-3x+1$               
    \*                    $(x^2 + x + 1)  (x^8 + x^5 + x^4 + x^3 + 1)$                       impossible
          10 2.04817...   $x^{10}-x^9-2x^8-x^5-2x^2-x+1$                                     
    \*                    $x^{10} + x^9 + x^5 + x + 1$                                       impossible
    36     8 2.04952...   $x^8-x^7-x^6-x^5-2x^4-x^3-x^2-x+1$                                 
    \*                    $(x + 1)^4  (x^4 + x^3 + x^2 + x + 1)$                             
          10 2.05286...   $x^{10}-x^9-x^8-2x^7-x^5-2x^3-x^2-x+1$                             
    \*                    $(x^4 + x^3 + x^2 + x + 1)  (x^6 + x^3 + 1)$                       impossible (\*)
          10 2.05353...   $x^{10}-2x^9-x^8+2x^7-x^5+2x^3-x^2-2x+1$                           
    \*                    $(x^2 + x + 1)  (x^8 + x^7 + x^6 + x^4 + x^2 + x + 1)$             impossible
          10 2.05523...   $x^{10}-x^9-x^8-x^7-x^6-3x^5-x^4-x^3-x^2-x+1$                      
    \*                    $x^{10} + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1$   impossible (\*)
    37    10 2.05631...   $x^{10}-2x^9-x^7+x^6+x^5+x^4-x^3-2x+1$                             
    \*                    $(x^2 + x + 1)^3  (x^4 + x^3 + x^2 + x + 1)$                       
          10 2.05819...   $x^{10}-x^9-3x^7-x^6-3x^5-x^4-3x^3-x+1$                            
    \*                    $(x^5 + x^3 + x^2 + x + 1)  (x^5 + x^4 + x^3 + x^2 + 1)$           impossible
    38     8 2.06017...   $x^8-3x^7+2x^6+x^5-3x^4+x^3+2x^2-3x+1$                             
    \*                    $(x^4 + x + 1)  (x^4 + x^3 + 1)$                                   
          10 2.06226...   $x^{10}-2x^8-3x^7-2x^6-x^5-2x^4-3x^3-2x^2+1$                       
    \*                    $x^{10} + x^7 + x^5 + x^3 + 1$                                     impossible
          10 2.06420...   $x^{10}-2x^9+x^8-2x^7-x^5-2x^3+x^2-2x+1$                           
    \*                    $(x^2 + x + 1)  (x^8 + x^7 + x^6 + x^4 + x^2 + x + 1)$             impossible
          10 2.06557...   $x^{10}-x^8-3x^7-4x^6-5x^5-4x^4-3x^3-x^2+1$                        
    \*                    $(x^5 + x^2 + 1)  (x^5 + x^3 + 1)$                                 impossible
           8 2.06972...   $x^8-2x^7+x^5-3x^4+x^3-2x+1$                                       
    \*                    $x^8 + x^5 + x^4 + x^3 + 1$                                        impossible
          10 2.07416...   $x^{10}-3x^9+4x^8-6x^7+6x^6-7x^5+6x^4-6x^3+4x^2-3x+1$              
    \*                    $x^{10} + x^9 + x^5 + x + 1$                                       impossible
    39     4 2.08101...   $x^4-x^3-2x^2-x+1$                                                 
    \*                    $(x + 1)^2  (x^2 + x + 1)$                                         $\exists$ example
  ---- ----- ------------ ------------------------------------------------------------------ -------------------

[^1]: Y. M. is supported by JSPS KAKENHI Grant Numbers 15H05738 and 16K17560. H. O. is supported by JSPS KAKENHI 15K17521. S. R. is partially supported by National Science Centre, Poland, grant 2014/15/B/ST1/02197.

[^2]: There is a misprint in [@dolgachev16 Sect. 4.5, Table 2]: the terms $x^4$, $x^6$ appear with coefficient $6$ in $p_{f^*}(x)$. In we give the correct formula, but the misprint is irrelevant for us because we consider modulo 2 reduction.

[^3]: `https://arxiv.org/abs/1707.02563`
