2.1 Lattices

Let R be an integral domain of characteristic $0$ and K its field of fractions. We denote by $R^{\times }$ its group of units. In this paper, an R-lattice consists of a finitely generated projective R-module M and a nondegenerate, symmetric bilinear form $\langle \cdot , \colon \rangle M \times M \to K$ .

We call it integral if the bilinear form is R-valued, and we call it even if the square-norm of every element with respect to the bilinear form is in $2 R$ . If confusion is unlikely, we drop the bilinear form from notation and denote for $x,y \in M$ the value $\langle x, y \rangle $ by $xy$ and $\langle x, x \rangle $ by $x^2$ . The associated quadratic form is $Q(x) = x^2/2$ . We denote the dual lattice of M by $M^{\vee }$ . We call M unimodular if $M=M^{\vee }$ . For two lattices M and N, we denote by $M\perp N$ their orthogonal direct sum. The scale of M is $\operatorname {\mathrm {\mathfrak {s}}}(M)=\langle M, M \rangle $ , and its norm $\operatorname {\mathrm {\mathfrak {n}}}(L)$ is the fractional ideal generated by $\langle x, x \rangle $ for $x \in M$ . The set of self-isometries of M is the orthogonal group $O(M)$ of M.

We fix the following convention for the spinor norm: Let L be an R-lattice and $V = L \otimes K$ . Let $v \in V$ with $v^2 \neq 0$ . The reflection $\tau _v(x) = x - 2xv/v^2\cdot v$ is an isometry of V. The spinor norm of $\tau _v$ is defined to be $Q(v)=v^2/2 \in k^{\times }/(k^{\times })^2$ . By the Cartan–Dieudonné theorem $O(V)$ is generated by reflections. One can show that this defines a homomorphism $\operatorname {\mathrm {spin}}: O(V) \to k^{\times }/(k^{\times })^2$ by using the Clifford algebra of $(V,Q)$ .

An embedding $M \to L$ of lattices is said to be primitive if its cokernel is torsion-free. For $M\subseteq L$ , we denote by $M^{\perp L}=\{x \in L \mid \langle x, M \rangle =0\}$ the maximal submodule of L orthogonal to M. If confusion is unlikely, we denote it simply by $M^{\perp }$ . The minimum number of generators of a finitely generated R-module A will be denoted by $l(A)$ .

Let L be an even integral R-lattice. Its discriminant group is the group $D_L = L^{\vee }/ L$ equipped with the discriminant quadratic form $q_L\colon L \to K/2 R$ . Note that $l({D_{L}})\leq l(L)=\operatorname {\mathrm {rk}} L$ . Denote by $O(D_L)$ its orthogonal group, that is, the group of linear automorphisms preserving the discriminant form. If $f \colon L \to M$ is an isometry of even $\mathbb {Z}$ -lattices, then it induces an isomorphism $D_f \colon D_L \to D_M$ . Likewise, we obtain a natural map $O(L) \to O(D_L)$ , whose kernel is denoted by $O^{\sharp }(L)$ . For an isometry $f \in O(L)$ and H some subquotient of $L \otimes K$ preserved by f, we denote by $f|H$ the induced automorphism of H. Let $G\leq O(L)$ be a subgroup. We denote the fixed lattice by $L^G=\{x \in L \mid \forall g \in G \colon g(x)=x \}$ and its orthogonal complement by $L_G=(L^G)^{\perp }$ .

For $R=\mathbb {R}$ , let $s_+$ be the number of positive eigenvalues of a gram matrix and $s_-$ the number of negative eigenvalues. We call $(s_+,s_-)$ the signature pair or just signature of L.

2.2 Primitive extensions and glue

Let $R \in \{\mathbb {Z}, \mathbb {Z}_p\}$ and L be an even integral R-lattice. We call $M \perp N \subseteq L$ a primitive extension of $M \perp N$ if M and N are primitive in L and $\operatorname {\mathrm {rk}} L = \operatorname {\mathrm {rk}} M + \operatorname {\mathrm {rk}} N$ . Since L is integral, we have a chain of inclusions

$$\begin{align*}M \perp N \subseteq L \subseteq L^{\vee} \subseteq M^{\vee} \perp N^{\vee}.\end{align*}$$

The projection $M^{\vee } \perp N^{\vee } \to M^{\vee }$ induces a homomorphism $L/(M \perp N) \to D_M$ . This homomorphism is injective if and only if N is primitive in L. Let $H_M$ denote its image, and define $H_N$ analogously. The composition

$$\begin{align*}\phi \colon H_M \to L/(M \perp N) \to H_N\end{align*}$$

is called a glue map. It is an anti-isometry, that is, $q_M(x) = -q_N(\phi (x))$ for all $x \in H_M$ . Note that $L/(M\perp N) \leq H_M \perp H_N \leq D_M \perp D_N$ is the graph of $\phi \colon H_M \to H_N$ .

Conversely, any anti-isometry $\phi \colon H_M \to H_N$ between subgroups $H_M \subseteq D_M$ and $H_N \subseteq D_N$ is the glue map of a primitive extension: $M \perp N \subseteq L_{\phi }$ , where $L_{\phi }$ is defined by the property that $L_{\phi }/(M \perp N)$ is the graph of $\phi $ .

The determinants of the lattices in play are related as follows:

$$\begin{align*}\lvert\det L\rvert = \lvert D_M/H_M \rvert \cdot \lvert D_N/H_N \rvert = \lvert \det M \rvert \cdot \lvert \det N \rvert/[L : (M \perp N)]^2.\end{align*}$$

If $f_M \in O(M)$ and $f_N \in O(N)$ are isometries, then $g = f_M \oplus f_N$ preserves the primitive extension $L_{\phi }$ if and only if $\phi \circ D_{f_M} = D_{f_N} \circ \phi $ . We call $\phi $ an equivariant glue map with respect to $f_M$ and $f_N$ .

2.3 Lattices with isometry

We are interested in classifying conjugacy classes of isometries of a given $\mathbb {Z}$ -lattice. If the lattice in question is definite, its orthogonal group is finite. Using computer algebra systems, one can compute the group as well as representatives for its conjugacy classes. This approach breaks down if L is indefinite.

The unitary group $U(L)$ of the lattice with isometry L is the centralizer of $f_L$ in $O(L)$ . This is nothing but $\operatorname {\mathrm {Aut}}(L)$ in the category of lattices with isometry.

Terminology for lattices applies to lattices with isometry verbatim. For instance, the discriminant group $D_L$ of a lattice with isometry comes equipped with the induced isometry $D_{f_L} \in O(D_L)$ and $U(D_L)$ is the centralizer of $D_{f_L}$ in $O(D_L)$ . We denote by $G_L = \operatorname {Im}( U(L) \to U(D_L))$ .

Proposition 2.2. Let $M,N$ be lattices with isometry. Suppose that the characteristic polynomials of $f_M$ and $f_N$ are coprime. Then the double coset

$$\begin{align*}U(N) \backslash \{\mbox{equivariant glue maps } {D_{M}} \supseteq H_M \xrightarrow{\phi } H_N \subseteq D_{N} \} / U(M)\end{align*}$$

is in bijection with the set of isomorphism classes of lattices with isometry $(L,f)$ with characteristic polynomial $\chi _f(x) = \chi _{f_{M}}(x)\chi _{f_{N}}(x)$ and $M \cong (\ker \chi _{f_M}(f))$ and $N \cong (\ker \chi _{f_N}(f))$ .

3.1 $\mathbf {H}$ -markings

Let X be a K3 surface. We denote by

$$\begin{align*}\rho_X\colon \operatorname{\mathrm{Aut}}(X) \rightarrow O(\operatorname{\mathrm{H}}^2(X,\mathbb{Z})),\ g\mapsto (g^{-1})^*\end{align*}$$

the natural representation of the automorphism group $\operatorname {\mathrm {Aut}}(X)$ on the second integral cohomology group of X. It is faithful.

Let L be a fixed even unimodular lattice of signature $(3,19)$ . An L-marking of a K3 surface X is an isometry $\eta \colon \operatorname {\mathrm {H}}^2(X,\mathbb {Z}) \to L$ . The pair $(X, \eta )$ is called an L-marked K3-surface. A family of L-marked K3 surfaces is a family $\pi \colon \mathcal {X} \to B$ of K3 surfaces with an isomorphism of local systems $\eta \colon R^2\pi _*\underline {\mathbb {Z}} \to \underline {L}$ . If the base B is simply connected, then a marking of a single fiber extends to the whole family.

A family of H-marked K3 surfaces is a family of L-marked K3 surfaces $\pi \colon \mathcal {X} \to B$ of K3 surfaces with an isomorphism of local systems $\eta \colon R^2\pi _*\underline {\mathbb {Z}} \to \underline {L}$ and group of automorphisms $\mathcal {G} \leq \operatorname {\mathrm {Aut}}(\mathcal {X}/B)$ such that for each $b\in B$ the fiber $(\mathcal {X}_b,\eta _b,\mathcal {G}_b)$ is an H-marked K3 surface.

Let $(X,\eta ,G)$ be an H-marked K3 surface. The action of G on $\operatorname {\mathrm {H}}^{2,0}(X)$ induces via the marking $\eta $ a character $\chi \colon H \rightarrow \mathbb {C}^{\times }$ . We call such a character effective. We denote complex conjugation by a bar $\bar {\cdot }$ . Set $H_s = \ker \chi $ , and denote the $\chi $ -eigenspace by

$$\begin{align*}L_{\mathbb{C}}^{\chi} = \{x \in L \otimes \mathbb{C} \mid h(x) = \chi(h)\cdot x \mbox{ for all }h \in H\}.\end{align*}$$

Similarly let

$$\begin{align*}L_{\mathbb{R}}^{\chi+\bar{\chi}}=\{x \in L_{\mathbb{R}} \mid (h+h^{-1})(x) = \chi(h)x+\bar{\chi}(h)x \text{ for all } h \in H\}.\end{align*}$$

The generic transcendental lattice $T(\chi )$ is the smallest primitive sublattice of L such that $T(\chi ) \otimes \mathbb {C}$ contains $(L\otimes \mathbb {C})^{\chi }$ . We call $\operatorname {\mathrm {NS}}(\chi ) = T(\chi )^{\perp }$ the generic Néron–Severi lattice. Recall that $L_{H_s}$ is the complement of the fixed lattice $L^{H_s}$ .

Note that for any effective character $\chi \colon H \to \mathbb {C}^{\times }$ of H,

$$\begin{align*}\ker \chi = H_s = \{h \in H \mid L^h \mbox{ is of signature } (3,*)\}\end{align*}$$

is independent of $\chi $ . Indeed, because $L_{H_s}$ is negative definite, $L^{H_s}\subseteq L^h$ is of signature $(3,*)$ for any $h \in H_s$ . On the other hand, if $g\in H$ is not in $H_s$ , then $\chi (g)\neq 1$ and so $L^g$ is orthogonal to $L_{\mathbb {R}}^{\chi +\bar {\chi }}$ and contains $L^H$ . Therefore, $L^g$ is of signature $(1,*)$ . The kernel is a normal subgroup and $H/H_s \cong \mu _n$ is cyclic. We say that an effective group H is symplectic if $H=H_s$ and nonsymplectic otherwise.

In particular, this shows that $T(\chi )$ and $\operatorname {\mathrm {NS}}(\chi )$ do not depend on the choice of the effective character $\chi $ but only on H. We may denote them as $T(H)$ and $\operatorname {\mathrm {NS}}(H)$ .

3.2 Moduli spaces and periods

Let $\mathcal {M}_H$ denote the fine moduli space parametrizing isomorphism classes of H-marked K3 surfaces $(X,\eta ,G)$ . It is a non-Hausdorff complex manifold. It can be obtained by gluing the base spaces of the universal deformations of $(X,\eta ,G)$ ; see [Reference Brandhorst and Cattaneo19, §3].

Let $\chi \colon H \to \mathbb {C}^{\times }$ be an effective character. We have $\mathcal {M}_H = \mathcal {M}^{\chi }_H \cup \mathcal {M}^{\bar {\chi }}_H$ , where $\mathcal {M}_H^{\chi }$ parametrizes isomorphism classes of H-marked K3 surfaces $(X,\eta , G)$ with $\chi (\eta \rho _X(g)\eta ^{-1}) \cdot \omega _X = (g^*)^{-1} \omega _X$ for all $g \in G$ . By [Reference Brandhorst and Cattaneo19, Prop. 3.9], the forgetful map $\mathcal {M}_H \to \mathcal {M}_L$ , $(X,\eta ,G) \mapsto (X,\eta )$ into the moduli space of L-marked K3 surfaces is a closed embedding.

Definition 3.5. Let $H\leq O(L)$ be effective and $\chi \colon H \to \mathbb {C}^{\times }$ be a nontrivial effective character. We denote by

$$\begin{align*}\mathbb{D}^{\chi}= \{\mathbb{C} \omega \in \mathbb{P}((L\otimes \mathbb{C})^{\chi}) \mid \langle \omega, \omega \rangle=0, \langle \omega, \bar{\omega} \rangle>0\}\end{align*}$$

the corresponding period domain and its period map by

$$\begin{align*}\mathcal{P} \colon \mathcal{M}_H^{\chi} \to \mathbb{D}^{\chi}.\end{align*}$$

It is a local isomorphism (see, e.g., [Reference Brandhorst and Cattaneo19]). The discriminant locus is $\Delta = \bigcup \left \{\mathbb {P}(\delta ^{\perp }) \mid \delta \in L_H, \delta ^2=-2\right \} \subseteq \mathbb {P}(L_{\mathbb {C}})$ .

Denote by $N(H)=\{f \in O(L) \mid f H = H f\}$ the normalizer of H in $O(L)$ . If $(X,\eta ,G)$ is an H-marked K3 surface and $f \in N(H)$ , then $(X,f \circ \eta , G)$ is an H-marked K3 surface as well. In fact, all H-markings of $(X,G)$ arise in this way. So $N(H)$ is the group of changes of marking.

Set $\mathbb {D}_H = \mathbb {D}^{\chi } \cup \mathbb {D}^{\bar {\chi }}$ . The group $N(H)$ acts on $\mathbb {D}_H$ via an arithmetic subgroup of $O(T(H))$ , respectively $U(T(H))$ . Therefore, by [Reference Baily and Borel10] the space $\mathbb {D}_H/N(H)$ is a quasi-projective variety with only finite quotient singularities.

See [Reference Alexeev and Engel4, Reference Alexeev, Engel and Han5] for more on moduli of K3 surfaces and their compactifications.

3.3 Connected components

We next show that deformation classes of K3 surfaces with finite groups of automorphisms are precisely the connected components of the coarse moduli space of H-markable K3 surfaces $\mathcal {F}_H$ . In the following, for a topological space Y we denote by $\pi _0(Y)$ the set of (path) connected components of Y.

Let L be a K3 lattice and $H\leq O(L)$ an effective subgroup. Our next goal is to determine the connected components of the coarse moduli space $\mathcal {F}_H$ of H-markable K3 surfaces. Since the period domain $\mathbb {D}_H \setminus \Delta $ has exactly two connected components (Proposition 3.6 (3)), $\mathcal {F}_H$ has at most two components as well. It has only one connected component if and only if the action of $N(H)$ on $\mathbb {D}_H$ exchanges the two components.

Let $\chi : H \to \mathbb {C}^{\times }$ be an effective character. For $n \in N(H)$ , denote by $\chi ^n$ the character defined by $\chi ^n(h) = \chi (n^{-1} h n)$ . Denote by $N(\chi )$ the stabilizer of $\chi $ in $N(H)$ . For completeness sake, we mention the following proposition.

Let now $[H:H_s]=2$ . Then we have seen that $\chi =\bar {\chi }$ and $\mathbb {D}^{\chi }$ has two connected components. This can be dealt with by introducing positive sign structures. Our account follows [Reference Shimada72]. A period $\mathbb {C} \omega \in \mathbb {D}^{\chi }$ can be seen as an oriented, positive definite real $2$ -plane. Indeed, the two real vectors $\operatorname {Re} \omega , \operatorname {Im} \omega \in L_{\mathbb {R}}$ give an ordered (and thus oriented), orthogonal basis of a positive definite plane in L.

For $[H:H_s]>2$ , the periods in $\mathbb {D}^{\chi }$ all give the same sign structure. But for $[H:H_s]=2$ , there are two sign structures which give the two connected components of the period domain. Whether or not $N(H)$ preserves the sign structure is encoded by a certain character. See section 2.1 for our conventions on the spinor norm.

Let L be an $\mathbb {R}$ -lattice. We denote by $O^+(L)=\ker (\det \cdot \operatorname {\mathrm {spin}})$ the subgroup of orientation preserving isometries. If $G \leq O(L)$ is a subgroup, we denote by $G^+ = G \cap O^+(L)$ its normal subgroup of orientation preserving elements.

Proof. We claim that that $K \subseteq N_T$ . By definition any element $g\in K$ acts trivially on the discriminant group ${D_{T}}$ . Therefore, $g\oplus \operatorname {\mathrm {id}}_{\operatorname {\mathrm {NS}}(\chi )}$ extends to L. Since the restriction of any $h\in H$ to T is given by $\pm \operatorname {\mathrm {id}}_{T}$ , $g \oplus \operatorname {\mathrm {id}}_{\operatorname {\mathrm {NS}}(\chi )}$ commutes with h. The claim is proven. As a consequence, we have $N_T = {D_{}}^{-1}(J)$ . Thus, we have a commutative diagram with exact rows

where the vertical arrows are inclusions of normal subgroups. Hence, the cokernels exist and so we obtain the corresponding exact sequence

$$\begin{align*}1 \to K/K^+ \to N_T/N_T^+ \to J/J^+ \to 1\end{align*}$$

of the cokernels. To see this follow the proof of the snake lemma, which indeed is valid in this situation.

3.4 Saturated effective subgroups

By Theorem 3.9, the set of connected components of $\mathcal {F}_H$ is in bijection with the deformation classes of H-markable K3 surfaces. Our next goal is to enumerate all possible effective groups H up to conjugacy in $O(L)$ . The symplectic fixed and cofixed lattices $L^{H_s}$ and $L_{H_s}$ turn out to be the crucial invariants for this task. Let L be a $\mathbb {Z}$ -lattice and M a subset of $L \otimes \mathbb {C}$ . We denote by $O(L,M)=\{f \in O(L) \mid f_{\mathbb {C}}(M)=M\}$ the stabilizer of M.

3.5 Enumerating effective characters

Let $H_s \leq O(L)$ be a symplectic effective subgroup. We would like to enumerate conjugacy classes of effective characters $\chi \colon H \to \mathbb {C}^{\times }$ with the given heart $H_s$ .

Our next goal is to see how heart, head and spine determine the character. The first step is to make the definition of heart and head independent of a character.

By abuse of notation, we will identify $O(C)$ with $O(C)\times \{\operatorname {\mathrm {id}}_F\} \subseteq O(L\otimes \mathbb {Q})$ . Recall that for a glue map $\phi \colon {D_{F}} \to {D_{C}}$ , we denote by $F \perp C \subseteq L_{\phi }$ the corresponding primitive extension. Note that $L_{\phi }\cong L$ is a K3 lattice as well and $H_s$ preserves $L_{\phi }$ because all its elements act trivially on the discriminant group of C. Suppose that $\phi D_f \phi ^{-1} = D_c$ for some $c \in O(C)$ . Then $g = f \oplus c$ preserves $L_{\phi }$ . Set $H_{\phi } = \langle g , H_s \rangle $ . Since $H_s$ is saturated, any other choice of c is in $cH_s$ , so $H_{\phi }$ is independent of this choice. Let $\chi _{\phi }\colon H_{\phi } \to \mathbb {C}^{\times }$ be defined by $\chi (g)=\zeta _n$ .

If $L_1=L_2$ , then $\chi _1$ and $\chi _2$ are isomorphic if and only if they are conjugate. If moreover $H_1=H_2$ , then they are isomorphic if and only if they are conjugate by an element of $N(H)$ . Recall that $U(F,f)$ denotes the centralizer of f in $O(F)$ .

Theorem 3.25. Let $H_s$ be a heart, $(F,f)$ a head of $H_s$ and S the set of spines $\phi \colon {D_{F}} \to {D_{C}}$ . Then the double coset

$$\begin{align*}O(C) \backslash S / U(F,f)\end{align*}$$

is in bijection with the set of isomorphism classes of effective characters $\chi $ with the given heart and head.

4.1 Hermitian lattices and transfer

In this subsection, E is an étale $\mathbb {Q}$ -algebra with a $\mathbb {Q}$ -linear involution $\overline {\phantom {x}}\colon E \to E$ .

Let $\operatorname {\mathrm {Tr}} \colon E \to \mathbb {Q}$ be the trace. Note that E being étale is equivalent to the trace form $E\times E \to \mathbb {Q}, (x,y) \mapsto \operatorname {\mathrm {Tr}}(xy)$ being nondegenerate.

Note that an E-linear map preserves a Hermitian form if and only if it preserves the respective trace form. In view of these facts, we may work with E-bilinear forms and Hermitian forms over E interchangeably.

Given $x^n-1=\mu (x)\in \mathbb {Z}[x]$ and a lattice L, we seek to classify the conjugacy classes of isometries $f\in O(L)$ satisfying $\mu (f) = 0$ . To this aim, we put $E:=\mathbb {Q}[x,x^{-1}]/(\mu )\cong \mathbb {Q}[x]/(\mu )$ . This is an étale algebra and $x \mapsto x^{-1}$ defines an involution $\overline {\phantom {x}} $ on E. For $i \in I =: \{i \in \mathbb {N} \mid i \text { divides } n\}$ set $E_i=\mathbb {Q}[x]/(\Phi _i(x))$ . The algebra E splits as a direct product $E= \prod _{i \in I} E_i$ of cyclotomic fields with the induced involution. Let $(e_i)_{i \in I}$ be the corresponding system of primitive idempotents in E such that $\overline {e_i} = e_i$ and $E_i = E e_i$ .

Let $\Lambda = \mathbb {Z}[x]/(\mu )$ and $\Gamma = \prod _{i \in I} \Lambda e_i$ . The conductor of $\Gamma $ in $\Lambda $ is

$$\begin{align*}\mathfrak{f}_{\Gamma}= \{x \in \Lambda \mid \Gamma x \subseteq \Lambda\}.\end{align*}$$

It is the largest $\Gamma $ -ideal contained in $\Lambda $ . We obtain the following series of inclusions:

(4.1) $$ \begin{align} \mathfrak{f}_{\Gamma} \subseteq \Lambda \subseteq \Gamma = \prod_{i\in I} \Lambda e_i \end{align} $$

Example 4.5. Let $p \in \mathbb {Z}$ be a prime. For $\mu (x) = x^p-1 = (x-1)\Phi _p(x)$ , we obtain

$$\begin{align*}E = \mathbb{Q}[x] / (x^p-1) \cong \mathbb{Q}[x]/(x-1) \times \mathbb{Q}[x]/\Phi_p(x) \cong \mathbb{Q} \times \mathbb{Q}[\zeta_p].\end{align*}$$

Set $g(x) = \sum _{i=0}^{p-2}(i+1-p)x^i$ . One finds that $p = (x-1)g(x) + \Phi _p(x)$ . Hence, $e_1=\Phi _p(x)/p$ and $e_p = (x-1)g(x)/p$ . The conductor ideal is $\mathfrak {f}_{\Gamma }=p e_1 \Lambda + p e_p \Lambda $ . It contains p.

Let b be the bilinear form of the lattice L. A given isometry $f\in O(L,b)$ with minimal polynomial $\mu $ turns $(L,b)$ into a Hermitian $\Lambda $ -lattice $(L,h)$ by letting the class of x act as f. Note that for $x,y \in L\otimes E$ we have

$$\begin{align*}h(e_i x, e_j x) = e_i \bar e_j h(x,y) = e_i e_j h(x,y) = \delta_{ij} h(x,y).\end{align*}$$

Thus $e_i L$ is orthogonal to $e_j L$ for $i\neq j$ . Equation (4.1) yields the corresponding chain of finite index inclusions

(4.2) $$ \begin{align} \mathfrak{f}_{\Gamma}L \subseteq L \subseteq \Gamma L. \end{align} $$

Setting $L_i = e_i \mathfrak {f}_{\Gamma } L = L \cap e_i L = \ker \Phi _i(f)$ and $L_i' = e_i \Gamma L$ the outermost lattices are

Since $\Lambda e_i=\mathbb {Z}[x]/\Phi _i(x) = \mathbb {Z}_{E_i}$ is the maximal order in $E_i$ , $L_i$ and $L_i'$ are Hermitian lattices over the ring of integers of a number field. Such lattices are well understood. We use the outermost lattices of the sandwich to study the $\Lambda $ -lattice $(L,h)$ .

Example 4.6. Let L be a $\mathbb {Z}$ -lattice and $f\in O(L)$ an isometry of prime order p. Then $L_1 = \ker \Phi _1(f)$ and $L_p = \ker \Phi _p(f)$ . Since, by Example 4.5, $p \in \mathfrak {f}_{\Gamma }$ we have

(4.3) $$ \begin{align} p L \subseteq \mathfrak{f}_{\Gamma} L= L_1 \perp L_p \subseteq L. \end{align} $$

The idea for the classification is as follows: Given $\mu (x)$ and the $\mathbb {Z}$ -lattice $(L,b)$ , we get restrictions on the possible genera of the lattices $(L_i,h_i)$ from $(L,b)$ and the conductor. We take L as an overlattice of the orthogonal direct sum up to the action of the product of unitary groups $\prod _{i \in I}U(L_i,h_i)$ . Thanks to eq. (4.2), this is a finite problem. In practice, we successively take equivariant primitive extensions.

4.2 Glue estimates

The caveat of dealing with primitive extensions $A \perp B \subseteq C$ is that we do not know how to predict the genus of C. Or more precisely, how to enumerate all glue maps such that C lies in a given genus. So we have to resort to check this in line 19 of Algorithm 2 only after constructing C. To reduce the number of glue maps that have to be checked, in this section we prove various necessary conditions.

Proof. By Equation (4.3) $p C \subseteq A \perp B$ . Let $D_A \geq H_A \cong C/(A\perp B) =H\cong H_B \leq D_B$ be the glue between A and B. Note that these are isomorphisms as $\mathbb {Z}[x]$ -modules and $p H =0$ by Example 4.6.

The polynomial $\Phi _p$ annihilates B; hence, it annihilates $B^{\vee }$ and $H_B \leq D_B=B^{\vee }/B$ . Let $B' \leq p^{-1}B$ be defined by $B'/B = H_B$ . The $\mathbb {Z}[\zeta _p]$ -module $B'$ is a finitely generated torsion-free module of rank $e_p$ . By the invariant factor theorem over Dedekind domains [Reference Curtis and Reiner27, (22.13)], any torsion quotient module of $B'$ is generated by at most $e_p$ elements. On the other hand, since $H_B \leq D_B$ , it is generated by at most $l(D_B)$ elements as a $\mathbb {Z}$ -module, in particular as $\mathbb {Z}[x]$ -module.

Similarly, $H_A$ is annihilated by $\Phi _1$ and generated by at most $\min \{e_1,l(D_A)\}$ elements. Since the glue map is equivariant, the minimal number n of generators of H as $\mathbb {Z}[x]$ -module satisfies $n \leq m$ . As $H=C/(A\perp B)$ , viewed as a $\mathbb {Z}[\zeta _p]$ -module, is annihilated by the prime ideal P generated by $\Phi _1(\zeta _p)$ , we have $H\cong (\mathbb {Z}[\zeta _p]/P)^n$ . Since p is totally ramified in $\mathbb {Z}[\zeta _p]$ , the ideal P has norm p and thus

$$\begin{align*}[C:A\perp B]=|\mathbb{Z}[\zeta_p]/P|^n = p^n \mid p^m.\\[-38pt] \end{align*}$$

We call a torsion bilinear form $b \colon A \times A \to \mathbb {Q}_2/\mathbb {Z}_2$ evenif $b(x,x)=0$ for all $x \in A$ , otherwise we call it odd. Imitating [Reference Miranda and Morrison54, II §2], we define the functors $\rho _k$ .

Let be a Jordan decomposition with $f_{j}$ a unimodular bilinear form. Then one checks that $\rho _j(L) \cong (L_j/pL_j, \bar f_i)$ , where $\bar f_i$ is the composition of $p^{-1}f_i$ and the natural map $\mathbb {Q}_p \to \mathbb {Q}_p/\mathbb {Z}_p$ .

Let N be an integral lattice over $\mathbb {Z}_p$ and $l \in \mathbb {Z}$ such that $p^{l+1} N^{\vee } \subseteq N$ . Then $G_{l+1}(N)=G_{l+2}(N) = N^{\vee }$ and $G_{l}(N)=p^{-l}N \cap N^{\vee }$ . Using $pN^{\vee } \subseteq p^{-l}N$ , we obtain

$$\begin{align*}\rho_{l+1}(N) = N^{\vee} /(p^{-l}N \cap N^{\vee}). \end{align*}$$

Multiplication by $p^l$ gives the isomorphism

$$\begin{align*}\rho_{l+1}(N) \cong p^l N^{\vee}/ (N \cap p^l N^{\vee}) \cong p^lD_N.\end{align*}$$

Proof. Suppose that $p^l L^{\vee } \subseteq L$ . We prove (1)–(4).

Let $i \in \{1,2\}$ . By the assumptions,

$$\begin{align*}p^{l+1} L^{\vee} \subseteq p L \subseteq N_1 \perp N_2.\end{align*}$$

Since the extension is primitive, the orthogonal projection $\pi _i \colon L^{\vee } \rightarrow N_i^{\vee }$ is surjective. Applying $\pi _i$ to the chain of inclusions yields

$$\begin{align*}p^{l+1} N_i^{\vee} \subseteq p \pi_i(L) \subseteq N_i\end{align*}$$

which proves (1).

For (2), consider the inclusion $p^lL^{\vee } \subseteq L$ . A projection yields $p^l N_i^{\vee } \subseteq \pi _i(L)$ . Now (2) follows with $D_i = N_i^{\vee } / N_i$ and $H_i = \pi _i(L)/N_i$ .

(3) We have $\pi _i(p^l L^{\vee }) = p^l N_i^{\vee }$ . Recall that the glue map $\phi $ is defined by its graph $L/(N_1 \perp N_2)$ . Its subset

$$\begin{align*}p^l L^{\vee} / ((N_1 \perp N_2) \cap p^lL^{\vee})\end{align*}$$

projects onto both $p^l D_1$ and $p^l D_2$ . This proves the claim.

(4) Let $i\in {1,2}$ , $x_i \in N_1^{\vee }$ and $y_i \in N_2^{\vee }$ with $\hat \phi (\bar x_i) = \bar y_i$ , that is, $\phi (p^l x_i + N_1) = p^l y_i + N_2$ , that is,

$$\begin{align*}p^l(x_i+y_i) \in L.\end{align*}$$

In fact, from the proof of (3), we know a little more, namely that $p^l(x_i+y_i) \in p^l L^{\vee }$ so that $x_i+y_i \in L^{\vee }$ . This implies that $\langle p^l(x_1+y_1), x_2+y_2 \rangle \in \mathbb {Z}_p$ which results in

$$\begin{align*}b_{l+1}(\bar x_1, \bar x_2) \equiv p^{l}\langle x_1, x_2 \rangle \equiv - p^{l}\langle y_1, y_2 \rangle \equiv - b_{l+1}(\bar y_1, \bar y_2)\quad \mod \mathbb{Z}_p.\end{align*}$$

(4’) Suppose furthermore that both $\rho _{l+1}(N_i)$ are free and that $\rho _l(L)$ is even. Take $x=x_1=x_2 \in N_1^{\vee }$ and $y=y_1=y_2 \in N_2^{\vee }$ . Then $2^{l-1}\langle x+y, x+y \rangle {} \in \mathbb {Z}_2$ since $\rho _{l}(L)$ is even. Therefore, $2^{l}\langle x, x \rangle {} \equiv 2^l\langle y, y \rangle {}\ \ \mod 2 \mathbb {Z}$ .

Now, suppose that (1–4) hold for the glue map $\phi $ . Let $x + y \in L^{\vee }$ . We have to show that $p^l(x+y) \in L$ .

Let $w \in N_1^{\vee }$ , and $z \in N_2^{\vee }$ with $\hat \phi ( \bar w )=\bar z$ . By the definition of $\hat \phi $ and $\phi $ , this implies that $p^l(w+z) \in L$ . Therefore,

$$ \begin{align*} b_{l+1}(\bar y - \hat{\phi}(\bar x),\bar z) &= b_{l+1}( \bar y, \bar z) - b_{l+1}(\hat \phi (\bar x), \hat \phi(\bar w))\\ &= b_{l+1}( \bar y, \bar z) + b_{l+1}(\bar x, \bar w)\\ &\equiv p^l\langle y, z \rangle + p^{l}\langle x, w \rangle \\ &\equiv \langle x+y, p^l(w+z) \rangle\\ &\equiv 0\quad \mod \mathbb{Z}_p. \end{align*} $$

Since the bilinear form on $\rho _{l+1}(N_2)$ is nondegenerate, this shows that $\hat \phi (\bar x)=\bar y$ . By the definition of a glue map, we obtain $p^l(x+y) \in L$ .

Suppose furthermore that (4’) holds, so $p=2$ , $\rho _{l+1}(N_i)$ is free and $\hat \phi $ preserves the induced quadratic forms. Let $x+y \in L^{\vee }$ ; we have to show that $q_l(\overline {x+y})\equiv 0\ \ \mod \mathbb {Z}$ . Indeed,

$$ \begin{align*} 2q_l(\overline{x+y}) &= 2^{l} \langle x+y, x+y \rangle{}\\ &\equiv q_{l+1}(\bar x)+q_{l+1}(\bar y)\\ &\equiv q_{l+1}(\bar x)+q_{l+1}(\hat \phi(\bar x)) \\ &\equiv 0\quad \mod 2 \mathbb{Z}.\\[-34pt] \end{align*} $$

Note that $(C,0,C)$ and $(0,C,C)$ are p-admissible for all p. We call a triple $(\mathcal {A},\mathcal {B},\mathcal {C})$ of genera of $\mathbb {Z}$ -lattices p-admissible if for any representatives A of $\mathcal {A}$ , B of $\mathcal {B}$ and C of $\mathcal {C}$ the triple $(A,B,C)$ is p-admissible.

Remark 4.14. For the existence of the (not necessarily primitive!) embeddings in (6), there is a necessary and sufficient criterion found in [Reference O’Meara66, Theorem 3]. Note that condition (V) in said theorem is wrong. The correct condition is

$$\begin{align*}(\mathrm{V}) \qquad 2^i (1+4 \omega) \to (2^i \oplus \mathfrak{L}_{i+1})/ \mathfrak{l}_{[i]}.\end{align*}$$

Thus, being p-admissible is a condition that can be checked easily algorithmically.

4.3 Enumeration of conjugacy classes of isometries

Let $p\neq q$ be prime numbers. In this subsection, we give an algorithm which, for a given genus of $\mathbb {Z}$ -lattices $\mathcal {L}$ , computes a complete set of representatives for the isomorphism classes of lattices with isometry $(L,f)$ of order $p^i q^j$ such that L is in $\mathcal {L}$ .

Let $(L,f)$ be a lattice with isometry. As before, we will drop f from the notation and simply denote it by L and the corresponding isometry by $f_L$ . If $N \leq L$ is an f-invariant sublattice, we view it as a lattice with isometry $f_N = f|_N$ .

The data structure we use for lattices with isometry is a triple $(L,f_L,G_L)$ , where $G_L$ is the image of $U(L) \to O({D_{L}})$ and $U(L)$ denotes the centralizer of $f_L$ in $O(L)$ .

By abuse of terminology, we call such a triple a lattice with isometry as well. So every algorithm in this section which returns lattices with isometry actually returns such triples $(L,f_L,G_L)$ (or at least a function which is able to compute $G_L$ when needed). We omit $f_L$ and $G_L$ from notation and denote the triple simply by L.

Since we can encode a genus in terms of its symbol and can check for equivalence of two given symbols efficiently, the type is an effectively computable invariant.

Proof. Suppose that $A \perp B \subseteq C$ is an equivariant primitive extension with $pC \subseteq A \perp B$ and $t(C,f_C^p)=t(C')$ . Let $\phi \colon H_A \to H_B$ be the corresponding glue map. It is admissible by Proposition 4.10. The existence of g in line 2 follows from $(A,B,C')$ being p-admissible. Further, $p C \subseteq A \perp B$ , gives $H_A \subseteq (p^{-1}A\cap A^{\vee })/A$ .

Since $\phi $ is equivariant, we get that $\mu _B(G_{f_A})$ vanishes on $H_A$ . Hence, $H_A$ is a g-dimensional subspace of the $\mathbb {F}_p$ -vector space $V_A$ . It is stable under $f_A$ . Further, by Proposition 4.10, it contains $p^l D_A$ . Similarly, $H_B$ is preserved by $f_B$ , contains $p^l D_B$ and is contained in $V_B$ . Therefore, $(H_A,H_B)$ appears in the for loop in line 7.

Since $(A,B,C')$ is p-admissible, there exists an admissible glue map $\psi _0\colon H_A \to H_B$ . It can be computed using normal forms of quadratic or bilinear forms over finite fields. The set of admissible glue maps from $H_A$ to $H_B$ is given by $O(H_B, \rho _{l+1}(B))\psi _0$ . There exists an admissible equivariant glue map from $H_A$ to $H_B$ if and only if we find $g \in O(H_B,\rho _{l+1}(B))$ with

$$\begin{align*}g \psi_0 (f_A|H_A) = (f_B|H_B) g \psi_0.\end{align*}$$

Reordering we find $g \psi _0 (f_A|H_A) \psi _0^{-1} g^{-1} = f_B|H_B$ . This justifies lines 8 to 12 of the algorithm. So we continue with $\psi $ an equivariant admissible glue map. Now, the set of equivariant admissible glue maps is $O(H_B,\rho _{l+1}(B),f_B)\psi $ . Let $h \psi $ , $h \in O(H_B,\rho _{l+1}(B),f_B)$ be an equivariant admissible glue map, and let $a \in S_A$ and $b \in S_B$ . Then

$$\begin{align*}b h\psi a = (b|_{H_B}) h (\psi a \psi^{-1}) \psi = h' \psi.\end{align*}$$

Therefore, $S_B h\psi S_A \mapsto S_B h S_A^{\psi } \psi $ defines a bijection of

$$\begin{align*}S_B \backslash \{\mbox{equivariant admissible glue maps } \psi\colon H_A \to H_B\}/S_A \end{align*}$$

with the double coset

$$\begin{align*}S_B \backslash O(H_B,\rho_{l+1}(B),f_B)/S_A^{\psi}.\end{align*}$$

Finally, the condition on $(a,b)$ in the equation for $S_C$ in line 22 of the algorithm is indeed the one to preserve $C/(A\perp B) \leq {D_{A \perp B}}$ . Thus, $S_C$ is the stabilizer of $C/(A \perp B)$ in $G_A \times G_B$ .

Algorithm 3 relies on an enumeration of genera of Hermitian lattices over maximal orders of number fields with bounds on the determinant and level. Then for each genus a single representative is computed [Reference Kirschmer42, Algorithm 3.5.6] and its type is compared with that of A. Finally, Kneser’s neighbor method [Reference Kirschmer42, §5] is used to compute representatives for the isometry classes of the genus.

By calling NextP on a complete set of representatives of the types of lattices with isometry of order $p^dq^e$ , we can obtain a complete set of representatives for the isomorphism classes of lattices with isometry of order $p^{d+1}q^e$ . By iterating this process, we have an algorithm to enumerate representatives for all isomorphism classes of lattices with isometries of a given order $p^d q^e$ .

4.4 Computation of the group $G_L$

The algorithms of the previous section for enumerating isomorphism classes of lattices with isometry require as input for each lattice with isometry L the group $G_L$ , which is the image of the natural map $U(L) \to U(D_L)$ . Recall that we use a recursive approach and for primitive extensions C of lattices with isometry A and B, the group $G_C$ can be determined using $G_A$ and $G_B$ (see Algorithm 2). It is therefore sufficient to explain how $G_L$ can computed for the lattices constructed in Algorithm 3, which form the base case of the recursive strategy. We therefore consider lattices with isometry L such that $f_L$ has irreducible minimal polynomial and $\mathbb {Z}[f_L]$ is the maximal order of $\mathbb {Q}[f_L]$ . We distinguish the following four cases:

1. 1. The lattice L is definite. Then $O(L)$ is finite and can be computed using an algorithm of Plesken and Souvignier [Reference Plesken and Souvignier68].

2. 2. The lattice L is indefinite of rank $2$ and $f_L = \pm 1$ . For this situation, the computation of $G_L$ will be explained in the remainder of this section.

3. 3. The lattice L is indefinite of rank $\geq 3$ and $f_L = \pm 1$ . In this case, Miranda–Morison theory [Reference Miranda and Morrison55, Reference Miranda and Morrison56] along with some algorithms by Shimada [Reference Shimada72] solve the problem. A short account of this is given in Section 5.

4. 4. The automorphism satisfies $f_L \neq \pm 1$ . This will be addressed in Section 6, where we extend the theory of Miranda–Morison to the Hermitian case.

We end this section by describing the computation of $G_L$ in case (2). Therefore, let L be be an indefinite binary lattice over $\mathbb {Z}$ and $V = L \otimes \mathbb {Q}$ the ambient quadratic space of discriminant $d \in \mathbb {Q}^{\times }/(\mathbb {Q}^{\times })^2$ .

It follows from [Reference Eichler30, §5] that we may assume that V is a two-dimensional étale $\mathbb {Q}$ -algebra and $L \subseteq V$ is a $\mathbb {Z}$ -lattice of rank $2$ . More precisely, V is isomorphic to the Clifford algebra $C^+$ , which in turn is isomorphic to $\mathbb {Q}(\sqrt {d})$ . The $\mathbb {Q}$ -algebra V is a quadratic extension of $\mathbb {Q}$ if and only if d is not a square, which is the case if and only if V is anisotropic. If d is a square, then $V \cong \mathbb {Q} \times \mathbb {Q}$ . If $\sigma \colon V \to V$ denotes the nontrivial automorphism of V as a $\mathbb {Q}$ -algebra, then the quadratic form q on V is given by $q(x) = x \sigma (x)$ for $x \in V$ . Note that $\sigma \in O(V)$ and $\det (\sigma ) = -1$ .

Every element $y \in V$ induces an endomorphism $\tau _y \colon V \to V, \, x \mapsto yx$ of determinant $\det (\tau _y) = y \sigma (y) = q(y)$ . For a subset $X \subseteq V$ , set $X^1 = \{ x \in X \mid q(x) = 1 \}$ . The proper automorphism group $SO(V)$ of V is equal to $\{ \tau _y \mid y \in V^1\}$ and $O(V) = \{ \tau _y, \sigma \tau _y \mid y \in V^1 \}$ . We call two $\mathbb {Z}$ -lattices $I, J$ of V equivalent, if there exists $\alpha \in V^1$ such that $I = \alpha J$ . Finally, set

$$\begin{align*}\Lambda = \{ x \in V \mid xL \subseteq L \}, \end{align*}$$

which is a $\mathbb {Z}$ -order of V.