Hypergeometric groups and dynamics on K3 surfaces

Abstract

A hypergeometric group is a matrix group modeled on the monodromy group of a generalized hypergeometric differential equation. This article presents a fruitful interaction between the theory of hypergeometric groups and dynamics on K3 surfaces by showing that a certain class of hypergeometric groups and related lattices lead to a lot of K3 surface automorphisms of positive entropy, especially such automorphisms with Siegel disks.

Appendices

Calculations of resultants

We give a proof of the formulas in (31), which relies on the following lemma.

Lemma A.1

Let f(z) and g(z) be monic palindromic polynomials of even degrees. If F(w) and G(w) are the trace polynomials of f(z) and g(z) respectively, then $\mathrm {Res}(f, g) = \mathrm {Res}(F, G)^2$.

Proof

We have $f(z) = z^m F(z+ z^{-1})$ and $g(z) = z^n G(z+ z^{-1})$, where $\deg F(w) = m$ and $\deg G(w) = n$. Let $F(w) = \prod _{i=1}^m (w - A_i)$ and $G(w) = \prod _{j=1}^n (w - B_j)$. Set $\alpha _i + \alpha _i^{-1} = A_i$ and $\beta _j + \beta _j^{-1} = B_j$. We then have

$$\begin{aligned} f(z) = z^m \prod _{i=1}^m (z + z^{-1}-\alpha _i-\alpha _i^{-1}) = \prod _{i=1}^m \{ z^2 - (\alpha _i + \alpha _i^{-1}) z + 1\} = \prod _{i=1}^m (z-\alpha _i)(z-\alpha _i^{-1}), \end{aligned}$$

and similarly $g(z) = \prod _{j=1}^n (z-\beta _j)(z-\beta _j^{-1})$. Setting $\varDelta _{i j} := (\alpha _i - \beta _j) (\alpha _i - \beta _j^{-1}) (\alpha _i^{-1} - \beta _j) (\alpha _i^{-1} - \beta _j^{-1})$ we have

$$\begin{aligned} \varDelta _{i j}&= (\alpha _i-\beta _j) \cdot \dfrac{\alpha _i \beta _j -1}{\beta _j} \cdot \dfrac{1-\alpha _i \beta _j}{\alpha _i} \cdot \dfrac{\beta _j-\alpha _i}{\alpha _i \beta _j} = \left\{ \dfrac{(\alpha _i-\beta _j)(\alpha _i \beta _j -1)}{\alpha _i \beta _j} \right\} ^2, \\[1mm] A_i-B_j&= \alpha _i + \dfrac{1}{\alpha _i} - \beta _j - \dfrac{1}{\beta _j} = \alpha _i-\beta _j + \dfrac{\beta _j-\alpha _i}{\alpha _i \beta _j} = \dfrac{(\alpha _i-\beta _j)(\alpha _i \beta _j -1)}{\alpha _i \beta _j}, \end{aligned}$$

and hence $\varDelta _{i j} = (A_i - B_j)^2$. Thus $\mathrm {Res}(f, g) = \prod _{i=1}^m \prod _{j=1}^n \varDelta _{i j} = \prod _{i=1}^m \prod _{j=1}^n (A_i - B_j)^2 = \mathrm {Res}(F, G)^2$. $\square$

Proof of Formulas (31a) and (31b). We use the following properties of the resultant: if $\deg f = m$ and $\deg g = n$ then $\mathrm {Res}(g, f) = (-1)^{m n} \mathrm {Res}(f, g)$; $\mathrm {Res}(f_1 f_2, \, g) = \mathrm {Res}(f_1, g) \cdot \mathrm {Res}(f_2, g)$; $\mathrm {Res}(z-\alpha , g) = g(\alpha )$ for $\alpha \in {\mathbb {C}}$. Put $\varphi _1(z) := z^{N-1} \Phi (z+z^{-1})$ in (30a). Then it follows from these properties and Lemma A.1 that

$$\begin{aligned} \mathrm {Res}(\varphi , \psi ) = \psi (1) \cdot \psi (-1) \cdot \mathrm {Res}(\varphi _1, \psi ) = \Psi (2) \cdot (-1)^N \Psi (-2) \cdot \mathrm {Res}(\Phi , \Psi )^2, \end{aligned}$$

which verifies (31a). Similarly, putting $\varphi _1(z) := z^N \Phi (z+z^{-1})$ and $\psi _1(z) := z^N \Psi (z+z^{-1})$ in (30b) we have

$$\begin{aligned} \mathrm {Res}(\varphi , \psi )&= (z+1)|_{z=1} \cdot \psi _1(1) \cdot \varphi _1(-1) \cdot \mathrm {Res}(\varphi _1, \psi _1)\\&= 2 \cdot \Psi (2) \cdot (-1)^N \Phi (-2) \cdot \mathrm {Res}(\Phi , \Psi )^2, \end{aligned}$$

which proves (31b) as desired. $\square$

Saito’s indices

We define the indices $\nu _p(f)$ for $p \in X_0(f)$ and $\nu _C(f)$ for $C \in X_1(f)$ in Saito’s formula (69), where $X_0(f)$ is the set of all fixed points and $X_1(f)$ is the set of all irreducible fixed curves of f. We also introduce the decomposition $X_1(f) = X_{\mathrm {I}}(f) \amalg X_{\mathrm {II}}(f)$. These tasks are mostly ring-theoretic. For details see [24] and [14, §3].

Let $A := {\mathbb {C}}[\![z_1, z_2]\!]$ and $\mathfrak {m}$ be its maximal ideal. Let $\sigma : A \rightarrow A$ be a nontrivial ring-endomorphism continuous in $\mathfrak {m}$-adic topology. It can be expressed as $\sigma (z_i) = z_i + g \cdot h_i$, $i = 1, 2$, for some g, $h_1$, $h_2 \in A$, where g is nonzero and $h_1$, $h_2$ are coprime. Consider the ideals $\mathfrak {a} = (g)$ and $\mathfrak {b} := (h_1, h_2)$ in A. Since $h_1$ and $h_2$ are coprime,

$$\begin{aligned} \delta (\sigma ) := \dim _{{\mathbb {C}}} (A/\mathfrak {b}) < \infty . \end{aligned}$$

Let $\varLambda (\sigma )$ be the set of all prime ideals $\mathfrak {p}$ of height 1 in A that divides $\mathfrak {a}$. we define

$$\begin{aligned} \nu _{\mathfrak {p}}(\sigma ) := \max \{ m \in {\mathbb {Z}}_{\ge 1} : \mathfrak {a} \subset \mathfrak {p}^m \} \qquad \hbox {for each} \quad \mathfrak {p} \in \varLambda (\sigma ). \end{aligned}$$

(79)

Let $\kappa [\mathfrak {p}]$ be the normalization of the ring $A/\mathfrak {p}$. It is isomorphic to ${\mathbb {C}}[\![t]\!]$ for some prime element t. Let

$$\begin{aligned} {\hat{\Omega }}^1_{A/{\mathbb {C}}} := \varprojlim _{n} \Omega ^1_{A_n/{\mathbb {C}}}, \qquad {\hat{\Omega }}^1_{\kappa [\mathfrak {p}]/{\mathbb {C}}} := \varprojlim _{n} \Omega ^1_{\kappa [\mathfrak {p}]_n/{\mathbb {C}}}, \end{aligned}$$

where $A_n := A/\mathfrak {m}^n$ and $\kappa [\mathfrak {p}]_n := {\mathbb {C}}[\![t]\!]/(t)^n$. Then ${\hat{\Omega }}^1_{A/{\mathbb {C}}} = A \, d z_1 \oplus A \, d z_2$ and ${\hat{\Omega }}^1_{\kappa [\mathfrak {p}]/{\mathbb {C}}} = \kappa [\mathfrak {p}] \, d t$. Let

$$\begin{aligned} \tau _{\mathfrak {p}} : {\hat{\Omega }}^1_{A/{\mathbb {C}}} \rightarrow {\hat{\Omega }}^1_{\kappa (\mathfrak {p})/{\mathbb {C}}} := {\hat{\Omega }}^1_{\kappa [\mathfrak {p}]/{\mathbb {C}}} \otimes _{\kappa [\mathfrak {p}]} \kappa (\mathfrak {p}) \end{aligned}$$

be the homomorphism induced from the natural map $A \rightarrow \kappa (\mathfrak {p})$, where $\kappa (\mathfrak {p})$ is the quotient field of $\kappa [\mathfrak {p}]$. Put

$$\begin{aligned} \varLambda _{\mathrm {I}}(\sigma ) := \{ \mathfrak {p} \in \varLambda (\sigma ) : \tau _{\mathfrak {p}}(\varpi _{\sigma }) \ne 0\}, \qquad \varLambda _{\mathrm {II}}(\sigma ) := \{ \mathfrak {p} \in \varLambda (\sigma ) : \tau _{\mathfrak {p}}(\varpi _{\sigma }) = 0\}, \end{aligned}$$

where $\varpi _{\sigma } := h_2 \cdot d z_1 - h_1 \cdot d z_2 \in {\hat{\Omega }}^1_{A/{\mathbb {C}}}$. There exists a nonzero element $a \in \kappa [\mathfrak {p}]$ such that

$$\begin{aligned} \tau _{\mathfrak {p}}(\varpi _{\sigma }) = a \cdot d t \quad \hbox {if } \mathfrak {p} \in \varLambda _{\mathrm {I}}(\sigma ); \qquad \varpi _{\sigma } = a \cdot d u \quad \hbox {if } \mathfrak {p} \in \varLambda _{\mathrm {II}}(\sigma ), \end{aligned}$$

where $u \in A$ is a prime element such that $\mathfrak {p} = (u)$. Under the identification $\kappa [\mathfrak {p}] = {\mathbb {C}}[\![t]\!]$ we define

$$\begin{aligned} \mu _{\mathfrak {p}}(\sigma )&:= \max \{ m \in {\mathbb {Z}}_{\ge 0} : (a) \subset (t)^m \}, \qquad \hbox {for each} \quad \mathfrak {p} \in \varLambda (\sigma ), \nonumber \\[1mm] \nu _A(\sigma )&:= \delta (\sigma ) + \sum _{\mathfrak {p} \in \varLambda (\sigma )} \nu _{\mathfrak {p}}(\sigma ) \cdot \mu _{\mathfrak {p}}(\sigma ). \end{aligned}$$

(80)

Back in the geometric situation we first define $\nu _p(f)$ at $p \in X_0(f)$. The completion ${\hat{{\mathcal {O}}}}_{X, p}$ of the local ring ${\mathcal {O}}_{X, p}$ is identified with $A = {\mathbb {C}}[\![z_1, z_2]\!]$ and the map f induces a nontrivial ring-endomorphism $\sigma := f^*_p : A \rightarrow A$ continuous in $\mathfrak {m}$-adic topology. Then $\nu _p(f)$ is defined to be $\nu _A(\sigma )$ in (80). Next we define $\nu _C(f)$ for $C \in X_1(f)$ as follows. Take a point $p \in C$ and consider $\sigma := f^*_p : A \rightarrow A$ again upon identifying ${\hat{{\mathcal {O}}}}_{X, p}$ with A. Let $C_p$ be the germ of C at p and let $\varLambda (C_p)$ be the set of all prime ideals in A determined by the irreducible components of $C_p$. Then one has $\varLambda (C_p) \subset \varLambda (\sigma )$, so $\nu _{\mathfrak {p}}(\sigma )$ makes sense for any $\mathfrak {p} \in \varLambda (C_p)$ via (79). Moreover the value of $\nu _{\mathfrak {p}}(\sigma )$ is independent of $p \in C$ and $\mathfrak {p} \in \varLambda (C_p)$. This common value is just $\nu _C(f)$ by definition. We can also show that either $\varLambda (C_p) \subset \varLambda _{\mathrm {I}}(\sigma )$ or $\varLambda (C_p) \subset \varLambda _{\mathrm {II}}(\sigma )$ holds with this dichotomy independent of $p \in C$. We have $C \in X_{\mathrm {I}}(f)$ in the former case while $C \in X_{\mathrm {II}}(f)$ in the latter case. If C is smooth and transverse then $C \in X_{\mathrm {I}}(f)$.

Symbols for hypergeometric groups

There are a lot of symbols in the sections (Sects. 2–6) concerning hypergeometric groups. Here is a list of them with brief explanations including the places where they appear for the first time.

• H, hypergeometric group in $\mathrm {GL}(n, {\mathbb {C}})$, Sect. 2,

• A, B, matrices generating H, Sect. 2,

• $\varvec{a}= \{a_1, \dots , a_n \}$, eigenvalues of A, Sect. 2,

• $\varvec{b}= \{b_1, \dots , b_n\}$, eigenvalues of B, Sect. 2,

• $a^{\dagger } := {\bar{a}}^{-1}$, reciprocal of conjugate of $a \in {\mathbb {C}}^{\times }$, Sect. 2,

• $\varphi (z)$, characteristic polynomials of A, Sect. 2,

• $\psi (z)$, characteristic polynomials of B, Sect. 2,

• $\mathrm {Res}(\varphi , \psi )$, resultant of $\varphi (z)$ and $\psi (z)$, Sect. 2,

• $C := A^{-1}B$, complex reflection, $c := \det C$, Sect. 2,

• $\varvec{r}$, eigenvector of C corresponding to c, Sect. 2,

• $(\, \cdot \, , \, \cdot \,)$, H-invariant Hermitian form, Sect. 2,

• $\varvec{a}_{\mathrm {on}}$/$\varvec{a}_{\mathrm {off}}$, parts of $\varvec{a}$ lying on/off $S^1$, Sect. 3,

• $\varvec{b}_{\mathrm {on}}$/$\varvec{b}_{\mathrm {off}}$, parts of $\varvec{b}$ lying on/off $S^1$, Sect. 3,

• $\varvec{a}_1, \dots , \varvec{a}_t$, clusters in $\varvec{a}_{\mathrm {on}}$, Sect. 3,

• $\varvec{b}_1, \dots , \varvec{b}_t$, clusters in $\varvec{b}_{\mathrm {on}}$, Sect. 3,

• $[\varvec{a}_{\mathrm {on}}]$, $[\varvec{b}_{\mathrm {on}}]$, configurations of $\varvec{a}_{\mathrm {on}}$, $\varvec{b}_{\mathrm {on}}$, Sect. 3,

• $2 \pi \alpha _i$, $2 \pi \beta _i$, arguments of $a_i$, $b_i$, Sect. 3,

• $\gamma := \beta _1 + \cdots + \beta _n -(\alpha _1+\cdots +\alpha _n)$, Sect. 3,

• $\varepsilon = \pm 1$, signature of $\sin \pi \gamma$, Sect. 3.1,

• $p-q$, index of invariant Hermitian form, Sect. 3.1,

• $E(\lambda )$, generalized eigenspace for $\lambda \in \varvec{a}\cup \varvec{b}$, Sect. 3.3,

• $m(\lambda ) := \dim E(\lambda )$, multiplicity of $\lambda$, Sect. 3.3,

• $\mathrm {idx}(\lambda )$, local index at $\lambda \in \varvec{a}_{\mathrm {on}}\cup \varvec{b}_{\mathrm {on}}$, Sect. 3.3,

• $E(\mu , \mu ^{\dagger }) := E(\mu ) \oplus E(\mu ^{\dagger })$ for $\mu \in \varvec{a}_{\mathrm {off}}\cup \varvec{b}_{\mathrm {off}}$, Sect. 3.3,

• $\Phi (w)$, trace polynomials of $\varphi (z)$, Sect. 4.1,

• $\Psi (w)$, trace polynomials of $\psi (z)$, Sect. 4.1,

• $M(\lambda )$, multiplicity of $w -\lambda$ in $\Phi (w) \cdot \Psi (w)$, Sect. 4.1,

• $\varvec{A}$, $\varvec{B}$, multi-sets of roots of $\Phi (w)$, $\Psi (w)$, Sect. 4.2,

• $\varvec{A}_{\mathrm {on}}$/$\varvec{A}_{\mathrm {off}}$, parts of $\varvec{A}$ lying on/off $[-2, \, 2]$, Sect. 4.2,

• $\varvec{B}_{\mathrm {on}}$/$\varvec{B}_{\mathrm {off}}$, parts of $\varvec{B}$ lying on/off $[-2, \, 2]$, Sect. 4.2,

• $\varvec{A}_1, \dots , \varvec{A}_{s+1}$, trace clusters in $\varvec{A}_{\mathrm {on}}$, Sect. 4.2,

• $\varvec{B}_1, \dots , \varvec{B}_s$, trace clusters in $\varvec{B}_{\mathrm {on}}$, Sect. 4.2,

• $\varvec{A}_{\mathrm {in}}:= \varvec{A}_2 \cup \cdots \cup \varvec{A}_s$, Sect. 4.2,

• $\delta := (-1)^{ |\varvec{A}_{\mathrm {in}}| + |\varvec{B}_{\mathrm {on}} | +1}$ when n is even, Sect. 4.2,

• $\mathrm {Idx}(\tau )$, local index at $\tau \in \varvec{A}_{\mathrm {on}}\cup \varvec{B}_{\mathrm {on}}$, Sect. 4.3,

• $\varvec{A}_1^{\circ } := (\varvec{A}_1)_{<2}$, $\varvec{A}_{s+1}^{\circ } := (\varvec{A}_{s+1})_{>-2}$, Sect. 4.3,

• $\mathrm {Idx}({\mathcal {X}}) := \sum _{\tau \in {\mathcal {X}}} \mathrm {Idx}(\tau )$, local index on ${\mathcal {X}}$, Sect. 4.3,

• L, hypergeometric lattice, Sect. 5,

• $\mathrm {C}_k(z)$, kth cyclotomic polynomial, Sect. 5.2,

• $\mathrm {CT}_k(w)$, kth cyclotomic trace polynomial, Sect. 5.2,

• $\lambda _i$, Salem numbers from McMullen [18], Sect. 5.3,

• $S_i(z)$, minimal polynomial of $\lambda _i$, Sect. 5.3,

• $R_i(w)$, minimal trace polynomial of $\lambda _i$, Sect. 5.3,

• $\mathrm {L}(z)$, Lehmer’s polynomial, Sect. 5.3,

• $\mathrm {LT}(w)$, Lehmer’s trace polynomial, Sect. 5.3,

• $V(\lambda )$, $\lambda$-eigenspace of A in narrow sense, Sect. 6.2,

• $\tau = \tau (A)$, $\tau (B)$, special traces of A, B, Sect. 6.3.