Which Salem numbers occur as dynamical degrees of automorphisms of K3 surfaces: the degree-10 non-projective case and the projective characterization

Gross and McMullen (2002) asked which polynomials occur as characteristic polynomials of isometries of the indefinite even unimodular lattices, speculated that their three necessary conditions (reciprocity, signature parity, and three square conditions) are sufficient for monic irreducible polynomials, and proved the unramified case, deducing that every unramified Salem polynomial of degree 22 is the characteristic polynomial of an automorphism of a complex K3 surface of algebraic dimension 0. The lattice-theoretic questions they attached to this work, and the trace question for Salem numbers, have since been completely resolved. What remains open is the question raised by McMullen and studied systematically by Bayer-Fluckiger: determine exactly which Salem numbers occur as dynamical degrees of automorphisms of complex K3 surfaces. On surfaces of algebraic dimension 0, every Salem number of degree 4, 6, 8, 12, 14, 16 or 20 is realizable, degree 22 is characterized by the Gross-McMullen square conditions, and degrees 10 and 18 are governed by an explicit computable criterion that genuinely fails in degree 18 but is conjectured, on numerical evidence, to always hold in degree 10 - the last unresolved case on this side. For projective K3 surfaces no characterization is known at all. The problem asks for a complete characterization of the realizable Salem numbers in both settings.

Background

A lattice is a free abelian group $L$ of finite rank together with a symmetric bilinear form $(\, ,\,):L\times L\to\mathbb{Z}$ that is nondegenerate over $\mathbb{Q}$. It is called even if $(x,x)\equiv 0\pmod 2$ for every $x\in L$, and unimodular if the Gram matrix of the form in a $\mathbb{Z}$-basis has determinant $\pm 1$. Indefinite even unimodular lattices exist only in signatures $(p,q)$ with $p\equiv q\pmod 8$ and are then unique up to isometry; they are denoted $\mathrm{II}_{p,q}$. Gross and McMullen asked which polynomials arise as the characteristic polynomial of an isometry $F\in\mathrm{SO}(\mathrm{II}_{p,q})$, the group of determinant-one isometries. They proved that such a polynomial $S(x)\in\mathbb{Z}[x]$ must be reciprocal, i.e. $S(x)=x^{\deg S}S(x^{-1})$; that if exactly $2s$ of its roots counted with multiplicity lie off the unit circle, then $(p,q)\geq(s,s)$ componentwise and $(p,q)\equiv(s,s)\pmod 2$; and, via the spinor norm (a classical homomorphism from the special orthogonal group of a rational quadratic space to the square classes $\mathbb{Q}^\times/\mathbb{Q}^{\times2}$), that $|S(1)|$, $|S(-1)|$ and $(-1)^nS(1)S(-1)$ must all be perfect squares, where $2n=\deg S=p+q$ (Theorem 6.1, stated for separable $S$). They speculated that for monic irreducible $S$ these conditions are also sufficient, proved sufficiency in the unramified case $|S(1)|=|S(-1)|=1$ by realizing $S$ as the characteristic polynomial of the multiplication-by-$x$ map on an ideal lattice $\mathrm{Tr}_{K/\mathbb{Q}}(\xi \alpha\beta)$ built from fractional ideals of the number field $K=\mathbb{Q}[x]/S(x)$ over its totally real subfield $k$ (unramified here means that the extension $K/k$ is unramified at all finite primes, which holds exactly when $|S(\pm 1)|=1$; unramifiedness at the primes above 2 was noted to suffice for evenness of the ideal lattice), and gave a reducible degree-10 polynomial satisfying the naive local conditions that nevertheless occurs for no $F\in\mathrm{SO}(\mathrm{II}_{9,1})$ (Proposition 5.2), showing that reducible polynomials need extra conditions.

A Salem number is a real algebraic integer $\alpha>1$ whose minimal polynomial - its Salem polynomial - is monic, irreducible and reciprocal and has exactly two roots off the unit circle, namely $\alpha$ and $\alpha^{-1}$; all other roots lie on the unit circle. The degree is even and at least 4 in the modern convention followed here; Gross and McMullen additionally admit degree 2, i.e. reciprocal quadratic integers, but that case is excluded from the problem below. The smallest known Salem number is Lehmer's number $\lambda_{10}\approx 1.17628$, of degree 10. Cyclotomic polynomials $\Phi_m$ are the minimal polynomials of the roots of unity. Gross and McMullen proved (Theorem 1.6) that infinitely many unramified Salem polynomials exist in every degree $2n$ with $n\geq 3$ odd - in particular in degree 22, the degree relevant for K3 surfaces, so the unramified case of their construction is not vacuous. (No unramified Salem polynomial exists in degree $2n$ with $n$ even: a Salem polynomial always satisfies $S(1)<0<S(-1)$, while unramifiedness requires $S(1)S(-1)=(-1)^n$.)

A complex K3 surface is a simply connected compact complex surface carrying a nowhere-vanishing holomorphic 2-form $\omega_X$; every complex K3 surface is Kaehler. Its middle cohomology $H^2(X,\mathbb{Z})$ with the cup-product pairing is an even unimodular lattice of signature $(3,19)$, hence isometric to $\mathrm{II}_{3,19}$. An automorphism $T$ of $X$ acts by an isometry $T^*$ of this lattice preserving the Hodge decomposition $H^2(X,\mathbb{C})=H^{2,0}\oplus H^{1,1}\oplus H^{0,2}$, and since $H^{2,0}$ is one-dimensional, $T^*$ multiplies $\omega_X$ by a scalar $\omega(T)$ with $|\omega(T)|=1$. The dynamical degree of $T$ is the spectral radius of $T^*$ on $H^2(X,\mathbb{Z})$, and its logarithm is the topological entropy. McMullen proved that the characteristic polynomial of $T^*$ is the product of at most one Salem polynomial and finitely many cyclotomic polynomials, so the dynamical degree is either 1 or a Salem number of degree at most 22. If $X$ is projective then $\omega(T)$ is a root of unity and the Salem factor has degree at most the Picard number $\rho(X)$ (the rank of the Neron-Severi lattice of divisor classes), hence at most 20; if $\omega(T)$ is a non-torsion root of the Salem polynomial then $X$ has algebraic dimension 0, meaning its field of meromorphic functions is $\mathbb{C}$. The strong Torelli theorem for K3 surfaces, combined with surjectivity of the period map, reduces the construction of automorphisms to the construction of isometries of $\mathrm{II}_{3,19}$ satisfying an explicit realizability condition - a signature condition on the part corresponding to $H^{2,0}\oplus H^{0,2}$, plus positivity of the induced isometry on its orthogonal complement (the realizable isometry criterion of McMullen, formulated as Theorem 6.2 of his 2011 paper).

Gross and McMullen applied this to prove that every unramified Salem polynomial of degree 22 is the characteristic polynomial of an automorphism of a complex K3 surface, necessarily of algebraic dimension 0, and that unramified degree-22 Salem numbers have trace at least $-2$, by the Lefschetz fixed point formula. McMullen's companion 2002 paper formulated the driving question - a first step toward determining which Salem numbers can arise as dynamical degrees - and built the first examples of K3 automorphisms with Siegel disks (domains on which the automorphism acts by an irrational rotation) from Salem numbers of degree 22 and trace $-1$ (the trace of a Salem number is the sum of its conjugates, i.e. minus the next-to-leading coefficient of its Salem polynomial); he subsequently realized Lehmer's number first on a non-projective K3 surface (2011) and then on a projective one (2016). The question 'Which Salem numbers occur as dynamical degrees of automorphisms of complex K3 surfaces?' is stated as Question 1 of Bayer-Fluckiger's 2025 paper, attributed there to McMullen.

On the lattice side the Gross-McMullen program is complete. Bayer-Fluckiger and Taelman confirmed the speculation for powers of monic irreducible polynomials, including the degree-22 consequence relevant to K3 surfaces. Bayer-Fluckiger then characterized, for polynomials without linear factors, exactly which occur, in terms of an obstruction group $X_F$ and Hasse principles; the Hasse principle genuinely fails in the reducible case (an explicit $\Phi_7^2\Phi_{14}^2$ example occurs everywhere locally but not globally on a definite rank-24 lattice). With Takada's subsequent work the characterization of characteristic polynomials of semisimple isometries of $\mathrm{II}_{p,q}$ - allowing linear factors and determinant $-1$ - is complete and computable; since the constructions produce semisimple isometries, this settles the existence form of the original lattice question. On the trace side, Smyth constructed infinitely many Salem numbers of trace $-1$ and McKee and Smyth proved that Salem numbers of every trace exist.

On the K3 side the question remains open in two places, and the two halves behave independently. For surfaces of algebraic dimension 0: all Salem numbers of degrees 4, 6, 8, 12, 14, 16 and 20 are realizable; degree 22 is characterized by the square conditions; degrees 10 and 18 are governed by an explicit computable criterion that genuinely fails in degree 18 and is expected to always hold in degree 10. For projective surfaces no characterization is known - only realized examples, necessary lattice conditions, and computer-aided criteria for restricted families. The smallest degree-18 Salem number is realizable projectively but not on surfaces of algebraic dimension 0, while certain degree-20 Salem numbers are realizable non-projectively but not projectively, so neither half implies the other.

Problem Statement

McMullen's question, stated as Question 1 of Bayer-Fluckiger (2025, arXiv:2209.06698) and attributed there to McMullen: determine exactly which Salem numbers occur as dynamical degrees of automorphisms of complex K3 surfaces. Since the characteristic polynomial of the action on $H^2(X,\mathbb{Z})$ contains at most one Salem factor, only even degrees $4,6,\ldots,22$ can occur, and on projective surfaces at most degree 20. Throughout, $\lambda_d$ denotes the smallest Salem number of degree $d$. Two components remain open.

(1) Degree 10 on surfaces of algebraic dimension 0 - the last open case on this side. Decide whether every Salem number of degree 10 is realizable, i.e. is the dynamical degree of an automorphism of a complex K3 surface of algebraic dimension 0. By Takada's criterion (2025, Theorem 1.5), this is equivalent to: for every Salem polynomial $S$ of degree 10 there exists $l$ in the finite set $C_{10}=\{1,2,3,4,5,6,7,8,9,10,11,12,14,15,16,18,21,22,24,28,30,36,42\}$ such that $S$ and the cyclotomic polynomial $\Phi_l$ have, after reduction modulo some prime $p$, a common irreducible factor of even degree that is symmetric (i.e. reciprocal: fixed up to sign by $x\mapsto x^{-1}$). This condition either holds for every degree-10 Salem polynomial, or fails for a specific degree-10 Salem polynomial, which is then not the Salem factor of the characteristic polynomial of the action on $H^2$ of any automorphism of a complex K3 surface of algebraic dimension 0.

(2) Projective characterization. Give necessary and sufficient conditions, expressed in terms of the Salem polynomial $S$, for a Salem number of even degree $4\leq d\leq 20$ to be the dynamical degree of an automorphism of a projective K3 surface - equivalently, to occur as the spectral radius of an integral isometry of a hyperbolic Neron-Severi lattice (the lattice of divisor classes with the intersection form, of signature $(1,\rho-1)$) that extends, together with a suitable isometry of its orthogonal complement in $\mathrm{II}_{3,19}$ (the transcendental lattice), to an isometry of $\mathrm{II}_{3,19}$ satisfying McMullen's Hodge-theoretic realizability condition, with $\omega(T)$ a root of unity. Known instances and non-instances: projectively realizable are Lehmer's number $\lambda_{10}$ and $\lambda_{18}$ (McMullen 2016), $\lambda_6$ (McMullen 2011) and $\lambda_{12}$ (Brandhorst and Gonzalez-Alonso 2018); not projectively realizable are $\lambda_{14}$, $\lambda_{16}$, $\lambda_{20}$ and the degree-20 Salem numbers of Bayer-Fluckiger (2025, Corollary 26.5).

The problem is completely solved when both components are settled.

References

1. Benedict H. Gross and Curtis T. McMullen, Automorphisms of even unimodular lattices and unramified Salem numbers, Journal of Algebra 257 (2002), no. 2, 265-290, DOI 10.1016/S0021-8693(02)00552-5, https://doi.org/10.1016/S0021-8693(02)00552-5.
2. Curtis T. McMullen, Dynamics on K3 surfaces: Salem numbers and Siegel disks, Journal fuer die reine und angewandte Mathematik 545 (2002), 201-233, DOI 10.1515/crll.2002.036, https://doi.org/10.1515/crll.2002.036.
3. Curtis T. McMullen, K3 surfaces, entropy and glue, Journal fuer die reine und angewandte Mathematik 658 (2011), 1-25, DOI 10.1515/crelle.2011.048, https://doi.org/10.1515/crelle.2011.048.
4. Curtis T. McMullen, Automorphisms of projective K3 surfaces with minimum entropy, Inventiones Mathematicae 203 (2016), no. 1, 179-215, DOI 10.1007/s00222-015-0590-z, https://doi.org/10.1007/s00222-015-0590-z.
5. Eva Bayer-Fluckiger and Lenny Taelman, Automorphisms of even unimodular lattices and equivariant Witt groups, Journal of the European Mathematical Society 22 (2020), no. 11, 3467-3490, DOI 10.4171/jems/991, https://doi.org/10.4171/jems/991, arXiv:1708.05540.
6. Eva Bayer-Fluckiger, Isometries of lattices and Hasse principles, Journal of the European Mathematical Society 26 (2024), no. 9, 3365-3428, DOI 10.4171/jems/1334, https://doi.org/10.4171/jems/1334, arXiv:2001.07094.
7. Eva Bayer-Fluckiger, Isometries of lattices and automorphisms of K3 surfaces, arXiv:2107.07583, https://arxiv.org/abs/2107.07583.
8. Eva Bayer-Fluckiger, Automorphisms of K3 surfaces, signatures, and isometries of lattices, Journal of the European Mathematical Society (2025), DOI 10.4171/jems/1598, https://doi.org/10.4171/jems/1598, arXiv:2209.06698.
9. Yuta Takada, Lattice isometries and K3 surface automorphisms: Salem numbers of degree 20, Journal of Number Theory 252 (2023), 195-242, DOI 10.1016/j.jnt.2023.05.006, https://doi.org/10.1016/j.jnt.2023.05.006, arXiv:2210.12946.
10. Yuta Takada, Characteristic polynomials of isometries of even unimodular lattices, Acta Arithmetica 217 (2025), no. 4, 333-378, DOI 10.4064/aa240212-2-9, https://doi.org/10.4064/aa240212-2-9, arXiv:2401.12620.
11. Eva Bayer-Fluckiger, Bert van Geemen and Matthias Schuett, Non-projective K3 surfaces with real or Salem multiplication, arXiv:2511.19970, https://arxiv.org/abs/2511.19970.
12. C. J. Smyth, Salem numbers of negative trace, Mathematics of Computation 69 (2000), no. 230, 827-838, DOI 10.1090/S0025-5718-99-01099-6, https://doi.org/10.1090/S0025-5718-99-01099-6.
13. James McKee and C. J. Smyth, There are Salem numbers of every trace, Bulletin of the London Mathematical Society 37 (2005), no. 1, 25-36, DOI 10.1112/S0024609304003790, https://doi.org/10.1112/S0024609304003790, arXiv:math/0308213.
14. Keiji Oguiso, The third smallest Salem number in automorphisms of K3 surfaces, in: Algebraic Geometry in East Asia - Seoul 2008, Advanced Studies in Pure Mathematics 60, Mathematical Society of Japan, Tokyo, 2010, 331-360, DOI 10.2969/aspm/06010331, https://doi.org/10.2969/aspm/06010331, arXiv:0905.2396.
15. Paul Reschke, Salem numbers and automorphisms of complex surfaces, Mathematical Research Letters 19 (2012), no. 2, 475-482, DOI 10.4310/mrl.2012.v19.n2.a18, https://doi.org/10.4310/mrl.2012.v19.n2.a18, arXiv:1202.5245.
16. Simon Brandhorst and Victor Gonzalez-Alonso, Automorphisms of minimal entropy on supersingular K3 surfaces, Journal of the London Mathematical Society 97 (2018), no. 2, 282-305, DOI 10.1112/jlms.12109, https://doi.org/10.1112/jlms.12109, arXiv:1609.02716.
17. Hélène Esnault, Keiji Oguiso and Xun Yu, Automorphisms of elliptic K3 surfaces and Salem numbers of maximal degree, Algebraic Geometry 3 (2016), no. 4, 496-507, DOI 10.14231/ag-2016-023, https://doi.org/10.14231/ag-2016-023, arXiv:1411.0769.
18. Simon Brandhorst, On the stable dynamical spectrum of complex surfaces, Mathematische Annalen 377 (2020), no. 1-2, 421-434, DOI 10.1007/s00208-020-01978-7, https://doi.org/10.1007/s00208-020-01978-7, arXiv:1708.06169.
19. Yuta Takada, Dynamical degrees of automorphisms of K3 surfaces with Picard number 2, Communications in Algebra 54 (2026), no. 4, 1772-1780, DOI 10.1080/00927872.2025.2565688, https://doi.org/10.1080/00927872.2025.2565688, arXiv:2405.15195.