ENRIQUES INVOLUTIONS AND BRAUER CLASSES

Abstract

We prove that every element of order 2 in the Brauer group of a complex Kummer surface X descends to an Enriques quotient of X. In generic cases, this gives a bijection between the set ${\mathcal Enr}(X)$ of Enriques quotients of X up to isomorphism and the set of Brauer classes of X of order 2. For some K3 surfaces of Picard rank $20,$ we prove that the fibers of ${\mathcal Enr}(X)\to \mathrm {{Br}}(X)[2]$ above the nonzero points have the same cardinality.

1 Introduction

Let S be a complex Enriques surface, and let $\pi \colon X\to S$ be its K3 étale double cover. J.-L. Colliot-Thélène asked whether the induced map of Brauer groups $\pi ^*\colon \mathrm {{Br}}(S)\simeq {\mathbb Z}/2\to \mathrm {{Br}}(X)$ is injective or zero ¹ . Beauville has given a necessary and sufficient condition for the injectivity of $\pi ^*$ [B, Cor. 5.7] and showed that the Enriques surfaces S for which this map is zero form a countable union of hypersurfaces in the moduli space of Enriques surfaces [B, Cor. 6.5]. Enriques surfaces with injective $\pi ^*$ are used in explicit constructions of Enriques surfaces over ${\mathbb Q}$ for which the Brauer–Manin obstruction fails to control weak approximation [HS1] and the Hasse principle [VV]). Enriques surfaces over ${\mathbb Q}$ such that the map $\pi ^*$ is zero have been constructed in [HS2, GS].

From a different perspective, one can start with a K3 surface X and consider the set $\mathcal F(X)\subset \mathrm {Aut}(X)$ of fixed point free involutions $\sigma \colon X\to X$ , which are precisely the involutions such that the quotient $X/\sigma $ is an Enriques surface.

In this paper, we are interested in the map

$$ \begin{align*} \Phi_X\colon{\mathcal F}(X) \longrightarrow \mathrm{{Br}}(X)[2], \end{align*} $$

which sends $\sigma \in \mathcal F(X)$ to $\pi ^*(b_S)$ , where $\pi \colon X\to X/\sigma =S$ is the quotient morphism, and $b_S$ is the unique nonzero element of $\mathrm {{Br}}(S)$ . A combination of results of Beauville and of Keum and Ohashi show that $\mathrm {{Im}}(\Phi _X)$ depends only on the isomorphism class of the transcendental lattice $T(X)$ of X (see Corollary 2.6). A description of all lattices $T(X)$ such that ${\mathcal F}(X)\neq \emptyset $ can be found in [BSV, Th. 1.6].

Let ${\mathcal Enr}(X)$ be the set of Enriques quotients of X, considered up to isomorphism of varieties. Equivalently, ${\mathcal Enr}(X)$ is the set of conjugacy classes of $\mathrm {Aut}(X)$ contained in $\mathcal F(X)$ (see [O1, Prop. 2.1]). Ohashi proved that the set ${\mathcal Enr}(X)$ is always finite [O1, Cor. 0.4] although its size is not bounded [O1, Th. 0.1]. The map $\Phi _X$ is $\mathrm { Aut}(X)$ -equivariant, where $\mathrm {Aut}(X)$ acts on $\mathcal F(X)$ by conjugation, so $\Phi _X$ descends to a map

$$ \begin{align*}\varphi_X\colon{\mathcal Enr}(X) \longrightarrow \mathrm{{Br}}(X)[2]/\mathrm{ Aut}(X).\\[-4pt]\end{align*} $$

The action of $\mathrm {Aut}(X)$ on $\mathrm {{Br}}(X)[2]$ factors through the action of the group of Hodge isometries of the integral Hodge structure on $T(X)$ , so when $\mathrm {Aut}_{\mathrm {Hdg}}(T(X))=\{\pm 1\}$ the action of $\mathrm {Aut}(X)$ on $\mathrm {{Br}}(X)[2]$ is trivial. In such a generic situation, $\varphi _X$ is a map ${\mathcal Enr}(X) \to \mathrm {{Br}}(X)[2]$ . In this case, the set ${\mathcal Enr}(X)$ depends only on the isomorphism class of the lattice $T(X)$ (see the discussion after Theorem 2.5).

Examples show that the set ${\mathcal Enr}(X)$ can be empty or very large, so in general $\varphi _X$ is neither surjective nor injective. A very general Enriques surface S (corresponding to the points of the moduli space outside a countable union of hypersurfaces) is the unique Enriques quotient of its K3 cover X; by Beauville, in this case, $\varphi _X({\mathcal Enr}(X))$ is a certain nonzero element of $\mathrm {{Br}}(X)[2]$ .

The aim of this paper is to clarify the structure of $\Phi _X$ and $\varphi _X$ in some favourable situations. Keum [K, Th. 2] proved that every Kummer surface is a double cover of some Enriques surface. His method can be used to prove the following.

Theorem A. Let X be a Kummer surface. Then, for every $\alpha \in \mathrm {{Br}}(X)$ of order $2,$ there is an Enriques quotient $\pi _S\colon X\to S$ such that $\alpha =\pi _S^*(b_S)$ .

In other words, for Kummer surfaces, the set $\mathrm {{Br}}(X)[2]\setminus \{0\}$ is contained in the image of $\Phi _X$ . As a kind of partial converse, in Corollary 2.7, we show that if X is a K3 surface such that the abelian group $\mathrm {{Br}}(X)[2]$ is generated by the image of $\Phi _X$ , then the transcendental lattice of X is divisible by 2 as an even lattice. We do not know if there exist Kummer surfaces such that $\Phi _X^{-1}(0)$ is non-empty. At the end of §2, we give examples of non-Kummer K3 surfaces such that $\mathrm {{Im}}(\Phi _X)=\{0\}$ .

In two generic cases, Ohashi classified all Enriques quotients of a given K3 surface. Combining Theorem A with his results [O1, Th. 4.1], [O2, Th. 1.1] we obtain the following corollary.

Corollary B. Let X be the Kummer surface attached to any of the following abelian surfaces:

1. (i) a product of two non-isogenous elliptic curves;

2. (ii) the Jacobian J of a curve of genus $2$ such that $\mathrm {NS\,}(J)\cong {\mathbb Z}$ .

Then $\varphi _X$ is a bijection between ${\mathcal Enr}(X)$ and $\mathrm {{Br}}(X)[2]\setminus \{0\}$ .

For some K3 surfaces of maximal Picard rank, the following result gives information about the fibers of $\varphi _X$ . Its proof uses a certain Galois action on $\mathrm {{Br}}(X)[2]$ constructed by the second named author in [V].

Theorem C. Let X be a K3 surface of Picard rank $20$ . Let $E={\mathbb Q}(\sqrt {-d})$ , where d is the discriminant of the transcendental lattice $T(X)$ . Assume that $\mathrm {{End}}_{\mathrm {Hdg}} (T(X))$ is the ring of integers ${\mathcal O}_E\subset E$ and, moreover, $2$ is inert in E and $E \neq {\mathbb Q}(\sqrt {-3})$ . Then $\mathrm {Aut}_{\mathrm {Hdg}} (T(X))=\{\pm 1\}$ and the fibers of $\varphi _X\colon {\mathcal Enr}(X) \to \mathrm {{Br}}(X)[2]$ above the nonzero points have the same cardinality.

The conditions in Theorem C are easy to check. Let

$$ \begin{align*} \left(\begin{array}{cc} 2a&b\\b&2c\end{array}\right) \end{align*} $$

be the Gram matrix of $T(X)$ , where $a,b,c\in {\mathbb Z}$ , so that $-d=b^2-4ac<0$ . Write $-d=f^2D$ , where $f\in {\mathbb Z}$ and D is the discriminant of E. By [V, Th. 3.2] we have $\mathrm {{End}}_{\mathrm {Hdg}} (T(X))={\mathcal O}_E$ if and only if $f=\mathrm {gcd}(a,b,c)$ . Next, $2$ is inert in E if and only if $D\equiv 5 \bmod 8$ . If f is odd, so that $-d\equiv 5 \bmod 8$ , we have ${\mathcal Enr}(X)=\emptyset $ by [S], so in this case, the fibers of $\varphi _X$ are empty. Using Theorem A, it is easy to see that for each $D\equiv 5 \bmod 8$ , $D\neq -3$ , there are infinitely many pairwise non-isomorphic K3 surfaces of Picard rank $20$ with complex multiplication by ${\mathcal O}_{{\mathbb Q}(\sqrt {D})}$ such that the fibers of $\varphi _X$ above the nonzero points of $\mathrm {{Br}}(X)[2]$ have the same positive number of elements.

It would be interesting to describe the K3 surfaces X such that $\Phi _X$ is surjective onto $\mathrm {{Br}}(X)[2]$ or onto $\mathrm {{Br}}(X)[2]\setminus \{0\}$ . In this direction, we have the following result, whose proof uses Nikulin’s theory of lattices [N] and surjectivity of the period map for K3 surfaces.

Theorem D. Let X be a K3 surface such that $\mathrm {{rk}}(\mathrm {NS\,}(X))\geq 12$ . Then there exist infinitely many K3 surfaces Y such that:

1. (1) $T(X)_{\mathbb Q} \cong T(Y)_{\mathbb Q}$ as polarized Hodge structures.

2. (2) The discriminants of $T(Y)$ are pairwise different.

3. (3) There is a natural isomorphism $\mathrm {{Br}}(X)[2]\cong \mathrm {{Br}}(Y)[2]$ under which

   $$ \begin{align*} \mathrm{{Im}}(\Phi_X)\setminus\{0\}=\mathrm{{Im}}(\Phi_Y)\setminus\{0\}. \end{align*} $$

We recall results of Beauville, Keum, and Ohashi, and then prove some useful lemmas in §2. Theorem A and Corollary B are proved in §3, Theorem C is proved in §4, and Theorem D in §5.

2 Lattices and the topology of Enriques quotients

A lattice L is a free finitely generated abelian group with a non-degenerate integral symmetric bilinear form. Write $L(2)$ for the same group with the form $2(x.y)$ .

For a lattice L, we denote by $A_L=L^*/L$ the discriminant group of L. If L is even, then $q_L\colon A_L\to {\mathbb Q}/2{\mathbb Z}$ is the associated quadratic form.

If $L\subset M$ are lattices, we denote by $L^{\perp }_M$ the orthogonal complement to L in M. It is clear that $L^{\perp }_M$ is a primitive sublattice of M.

Let $\mathrm {U}$ be the hyperbolic plane. Write $\mathrm {U}={\mathbb Z} e\oplus {\mathbb Z} f$ , where $(e^2)=(f^2)=0$ , $(e.f)=1$ . We denote by $\mathrm {E}_8$ the negative-definite, even, unimodular lattice of the root system $\mathrm { E}_8$ . Write

$$ \begin{align*} \Lambda=\mathrm{E}_8^{\oplus 2}\oplus \mathrm{U}^{\oplus 3}, \quad\quad M=\mathrm{U}(2)\oplus \mathrm{E}_8(2), \quad\quad N=\mathrm{U}\oplus \mathrm{U}(2)\oplus \mathrm{ E}_8(2). \end{align*} $$

Here, $\Lambda $ is the K3 lattice. Let $\iota \colon \Lambda \to \Lambda $ be the involution permuting two copies of $\mathrm {E}_8\oplus \mathrm {U}$ , and acting as $-1$ on the third copy of $\mathrm {U}$ . Then $\Lambda ^+\cong M$ and $\Lambda ^-\cong N$ , where $\Lambda ^{\pm }$ is the $\pm 1$ -eigenspace of $\iota $ . By [H2, (vii) on p. 305], for any Enriques quotient $\pi _S\colon X\to S=X/\sigma $ , the induced map

$$ \begin{align*} \pi_S\colon \mathrm{H}^2(S,{\mathbb Z})/_{\mathrm{tors}}\,\longrightarrow \mathrm{ H}^2(X,{\mathbb Z})\end{align*} $$

can be identified with the composition

$$ \begin{align*} \mathrm{H}^2(S,{\mathbb Z})/_{\mathrm{tors}}\simeq\mathrm{U}\oplus \mathrm{E}_8\stackrel{\mathrm{diag}}\longrightarrow (\mathrm{U}\oplus {\mathrm{E}}_8)^{\oplus 2}\subset \Lambda\simeq{\mathrm{H}}^2(X,{\mathbb Z}). \end{align*} $$

Here, the fixed point free involution $\sigma \colon X\to X$ induces the involution $\iota $ on $\Lambda $ .

The lattice N has a canonical character $N\to {\mathbb Z}/2$ which will play a crucial role in what follows.

Lemma 2.1. The homomorphism $\varepsilon \colon N\to {\mathbb Z}/2$ given by $\varepsilon (x):=\big (x.(e+f)\big )\bmod 2$ , where e and f are standard generators of $\mathrm {U}\subset N$ , does not depend on the embedding of lattices $\mathrm {U}\hookrightarrow N$ . Hence, $\alpha ^*(\varepsilon )=\varepsilon $ for any $\alpha \in \mathrm {Aut}(N)$ .

Proof. Let $e',f'$ be standard generators of $\mathrm {U}$ embedded in N. Write $e'=ae+bf+u$ , $f'=ce+df+w$ , where $a,b,c,d\in {\mathbb Z}$ and $u,w\in \mathrm {U}(2)\oplus \mathrm { E}_8(2)$ . We have $2ab+(u^2)=2cd+(w^2)=0$ and $ad+bc+(u.w)=1$ . Since $(u^2)$ and $(w^2)$ are divisible by 4, and $(u.w)$ is even, we see that $ab$ is even, $cd$ is even, and $ad+bc$ is odd. It follows that either $a,d$ are odd and $b,c$ are even, or $a,d$ are even and $b,c$ are odd. In both cases, $e'+f'$ equals $e+f$ modulo $2 \mathrm {U} \oplus \mathrm {U}(2)\oplus \mathrm {E}_8(2)$ , hence the result.

Lemma 2.2. If $x\in N$ is such that $(x^2)\equiv 2\bmod 4$ , then $\varepsilon (x)=0$ .

Proof. Write $x=ae+bf+u$ , where $a,b\in {\mathbb Z}$ and $u\in \mathrm {U}(2)\oplus \mathrm {E}_8(2)$ . Then a and b are both odd, hence $\varepsilon (x)\equiv a+b\equiv 0\bmod 2$ .

Lemma 2.3. Let L be a sublattice of N. If the restriction of $\varepsilon \colon N\to {\mathbb Z}/2$ to L is nonzero, then $L^{\perp }_N=L'(2)$ for some even lattice $L'$ .

Proof. Suppose $\varepsilon (x)\neq 0$ for some $x\in L$ . Writing $x=ae+bf+u$ , where $a,b\in {\mathbb Z}$ and $u\in \mathrm {U}(2)\oplus \mathrm {E}_8(2)$ , we see that a and b have opposite pairity. If $y=ce+df+w \in L^{\perp }_N$ , where $c,d\in {\mathbb Z}$ and $w\in \mathrm {U}(2)\oplus \mathrm {E}_8(2)$ , then $ad+bc$ is even, which implies that either c or d is even. Then $(y^2)=2cd +(w^2)$ is divisible by $4$ , hence $L^{\perp }_N=L'(2)$ for some even lattice $L'$ .

The importance of the character $\varepsilon \colon N\to {\mathbb Z}/2$ has been revealed by Beauville. Namely, let $\pi _S\colon X\to S=X/\sigma $ be an Enriques quotient of a K3 surface X. Let $T(X)\subset \Lambda $ be the transcendental lattice of X. Recall the canonical isomorphism

$$ \begin{align*} \mathrm{{Br}}(X)\cong\mathrm{{Hom}}(T(X),{\mathbb Q}/{\mathbb Z}) \end{align*} $$

(see [CS, (5.5) on p. 130, p. 142]). It is well known that the involution $\sigma $ is not symplectic [H2, Cor. 15.1.5 and (ii) on p. 356], so it acts on $\mathrm {H}^0(X,\Omega ^2_X)$ as $-1$ . Therefore, $\sigma ^*=\iota $ acts on $T(X)$ as $-1$ , so $T(X)\subset N$ .

Theorem 2.4 (Beauville).

Let $\pi _S\colon X\to S$ be an Enriques quotient of a K3 surface X. Then $\pi _S^*(b_S)\in \mathrm {{Br}}(X)[2]$ is the restriction of $\varepsilon \colon N\to {\mathbb Z}/2$ to $T(X)$ .

Proof. See [B, Prop s. 3.4 and 5.3].

An embedding $T(X)\subset N$ coming from an Enriques quotient of X is clearly primitive. The orthogonal complement $T(X)^{\perp }_N\subset N$ contains no $(-2)$ -elements x, because by Riemann–Roch either x or $-x$ is effective, but $\sigma ^*$ preserves effectivity. In fact, these are the only conditions. Horikawa’s theorem on the surjectivity of the period map for Enriques surfaces [H1] leads to the following result. See [K, Th. 1], which was extended in [O2, Prop. 2.1].

Theorem 2.5 (Keum, Ohashi).

Let X be a K3 surface. Associating to an Enriques quotient of X a primitive embedding $T(X)\subset N$ defines a bijection between ${\mathcal Enr}(X)$ and the set of equivalence classes of primitive embeddings of $T(X)$ into N without $(-2)$ -elements in the orthogonal complement. Here the embeddings $i_1$ and $i_2$ are equivalent if there is an automorphism $\tilde \phi $ of the lattice N and a $\phi \in \mathrm {Aut}_{\mathrm {Hdg}}(T(X))$ such that $i_2\circ \phi =\tilde \phi \circ i_1$ .

If $\mathrm {Aut}_{\mathrm {Hdg}}(T(X))=\{\pm 1\}$ (which holds, e.g., when the Picard number of X is odd), the set $\mathcal Enr(X)$ depends only on the lattice $T(X)$ .

Corollary 2.6. For any K3 surface $X,$ the following statements hold.

1. (i) $\mathrm {{Im}}(\Phi _X)\setminus \{0\}$ is the set of nonzero $\alpha \in \mathrm {{Br}}(X)[2]\cong \mathrm {{Hom}}(T(X),{\mathbb Z}/2)$ , for which there exists a primitive embedding $i\colon T(X)\hookrightarrow N$ such that $\alpha =i^*(\varepsilon )$ .

2. (ii) $0\in \mathrm {{Im}}(\Phi _X)$ if and only if there exists a primitive embedding $i\colon T(X)\hookrightarrow N$ without $(-2)$ -elements in the orthogonal complement such that $i^*(\varepsilon )=0$ .

3. (iii) If $x\in T(X)$ is such that $(x^2)\equiv 2\bmod 4$ , then $\alpha (x)=0$ for any $\alpha \in \mathrm {{Im}}(\Phi _X)$ .

Proof. Parts (i) and (ii) formally follow from Theorems 2.4 and 2.5 and Lemma 2.3. In particular, Lemma 2.3 implies that $i(T(X))^{\perp }_N$ does not contain $(-2)$ -classes. Part (iii) follows from Lemma 2.2.

Corollary 2.7. If X is a K3 surface such that the abelian group $\mathrm {{Br}}(X)[2]$ is generated by the image of $\Phi _X$ , then there is an even lattice $T'$ such that $T(X)\cong T'(2)$ .

Proof. It is enough to show that for every $x\in T(X)$ we have $(x^2)\equiv 0\bmod 4$ . Suppose that there is an element $y\in T(X)$ such that $(y^2)\equiv 2\bmod 4$ . Then y is not divisible by 2 in $T(X)$ . By Corollary 2.6(iii), the nonzero class of y in $T(X)/2T(X)$ is in the kernel of every $\alpha \in \mathrm {{Im}}(\Phi _X)$ . Thus $\mathrm {{Im}}(\Phi _X)$ is contained in a proper subgroup of $\mathrm {{Br}}(X)[2]$ .

Corollary 2.8. Let X be a K3 surface such that $T(X)$ has a basis $e_1,\ldots ,e_n$ with $(e_i^2)\equiv 2 \bmod 4$ for $i=1,\ldots ,n$ . Then either $\mathcal Enr(X)=\emptyset $ or $\mathrm {{Im}}(\Phi _X)=\{0\}$ .

Proof. Suppose that a nonzero $\alpha \in \mathrm {{Hom}}(T(X),{\mathbb Z}/2)$ is in the image of $\Phi _X$ . By Theorem 2.4, there is a primitive embedding $i\colon T(X)\to N$ such that $i^*(\varepsilon )=\alpha $ . By Lemma 2.2, we have $\alpha (e_i)=0$ for $i=1,\ldots ,n$ , hence $\alpha (T(X))=0$ which is a contradiction.

This can be used to give examples of K3 surfaces X such that $\mathrm {{Im}}(\Phi _X)=\{0\}$ . For example, one can take the K3 surface X of Picard rank 20 with transcendental lattice

$$ \begin{align*} \left(\begin{array}{cc}2&0\\ 0&2c\end{array}\right) \end{align*} $$

with $c=3, 5, 7$ . Indeed, by [SV, Table 3.1] in these cases, we have $|\mathcal Enr(X)|=1$ .

3 Kummer surfaces

Proof of Theorem A

By Corollary 2.6(i), it is enough to construct, for any nonzero $\alpha \in \mathrm {{Hom}}(T(X),{\mathbb Z}/2)$ , a primitive embedding $i\colon T(X)\hookrightarrow N=\mathrm {U}\oplus \mathrm {U}(2)\oplus \mathrm {E}_8(2)$ such that $\varepsilon (x)=\alpha (x)$ for any $x\in T(X)$ . We use Morrison’s classification of transcendental lattices of Kummer surfaces (see [H2, Cor. 14.3.20]). For each of them, Keum [K, pp. 106–108] constructed a primitive embedding into N; we follow the same strategy to construct all $2^n-1$ embeddings, where $n=\mathrm {{rk}}(T(X))$ . We keep the notation of [K], in particular, $e,f$ is a standard basis of $\mathrm {U}$ and $h,k$ is a standard basis of $\mathrm {U}(2)$ . We denote by $\rho $ the Picard rank of X.

In the proof below, we shall use the following particular case of a result of Nikulin.

Lemma 3.1. Any even negative-definite lattice of rank at most $4$ has a primitive embedding in $\mathrm {E}_8$ .

Proof. This follows from [N, Th. 1.12.4] using the fact that $\mathrm {E}_8$ is a unique even unimodular negative-definite lattice of rank 8.

$\rho =20$

In this case, the lattice $T={\mathbb Z} x\oplus {\mathbb Z} y$ is positive-definite with Gram matrix

$$ \begin{align*} \left(\begin{array}{cc}4a&2b\\2b&4c \end{array}\right),\end{align*} $$

where $a,b,c\in {\mathbb Z}$ . The three primitive embeddings can be given by sending $x,y$ to the following two elements of N:

$$ \begin{align*} (e+2af,2bf+h+ck), \!\quad (2bf+h+ak,e+2cf), \!\quad (e+2af, e+(2b-2a)f+h+(c-b+a)k). \end{align*} $$

$\rho =19$

Now T has signature $(2,1)$ . We can choose an integral basis $x,y,t$ of T so that the Gram matrix is

$$ \begin{align*}\left(\begin{array}{ccc}4a&2d&2l\\2d&4b&2m\\2l&2m&4c \end{array}\right),\end{align*} $$

where $a,b,c,d,l,m\in {\mathbb Z}$ and $a,b,c<0$ . The embeddings we need to construct are numbered by the nonzero vectors $(v_1,v_2,v_3)\in ({\mathbb F}_2)^3$ given by evaluating $\varepsilon $ on the images of $x,y,t$ in this order. By symmetry it is enough to construct embeddings labeled $(1,0,0)$ , $(1,1,0)$ , and $(1,1,1)$ . The first two can be given by sending $x,y,t$ to the following three elements of N, where w is a primitive element of $\mathrm {E}_8(2)$ such that $(w^2)=4c$ :

$$ \begin{align*} (e+2af,2df+h+bk,2lf+mk+w); \end{align*} $$

$$ \begin{align*} (e+2af,e+(2d-2a)f+h+(b-d+a)k,2lf+(m-l)k+w). \end{align*} $$

Next, we deal with $(1,1,1)$ . Without loss of generality, we can assume $m>0$ . Take

$$ \begin{align*} (e+k+ah,e+2mf+(d-m)h+w', e+lh+w), \end{align*} $$

where ${\mathbb Z} w'\oplus {\mathbb Z} w$ is a primitive sublattice of $\mathrm {E}_8(2)$ such that $(w^{\prime 2})=4b-4m<0$ , $(w^2)=4c<0$ , $(w'.w)=0$ .

$\rho =18$

Here, the lattice T is the orthogonal direct sum of ${\mathbb Z} x\oplus {\mathbb Z} y$ with signature $(1,1)$ and Gram matrix

$$ \begin{align*} \left(\begin{array}{cc}4a&2b\\2b&4c\end{array}\right) \end{align*} $$

and $\mathrm {U}(2)={\mathbb Z} r\oplus {\mathbb Z} s$ . Without loss of generality, we assume that $a,c<0$ and $b>0$ . Let w and u be primitive vectors of $\mathrm {E}_8(2)$ such that $(w^2)=4c<0$ and $(u^2)=4(a-b+c)<0$ . We label the embeddings in the same way as above. Up to exchanging the roles of x and y, and of r and s, it is enough to construct embeddings with the following labels:

$$ \begin{align*} (1,0,0,0), (1,1,0,0), (1,0,1,0), (0,0,1,0), (0,0,1,1), (1,1,1,0), (1,0,1,1), (1,1,1,1). \end{align*} $$

Let us first construct primitive embeddings with labels $(1,0,0,0)$ and $(1,1,0,0)$ by taking the direct sum of a primitive embedding ${\mathbb Z} x\oplus {\mathbb Z} y$ into $\mathrm { U}\oplus \mathrm {E}_8(2)$ and the identity embedding $\mathrm {U}(2)\tilde \longrightarrow \mathrm {U}(2)$ . We send $x,y$ to

$$ \begin{align*} (e+2af,2bf+w), \quad (e+2af,e+(2b-2a)f+u). \end{align*} $$

The embedding with label $(1,0,1,0)$ can be obtained by sending $x,y,r,s$ to

$$ \begin{align*} (e+2af-ak,2bf-bk+w,e+h,k). \end{align*} $$

For $(0,0,1,0)$ , we take $(h+w_1,bk+w_2,e,2e+2f+w_3)$ , where ${\mathbb Z} w_1\oplus {\mathbb Z} w_2\oplus {\mathbb Z} w_3$ is a primitive sublattice of $\mathrm {E}_8(2)$ with diagonal Gram matrix such that $(w_1^2)=4a<0$ , $(w_2^2)=4c<0$ , $(w_3^2)=-8$ .

For $(0,0,1,1)$ , we take $(h+w_1,bk+w_2,e,e+2f+w_3)$ , where ${\mathbb Z} w_1\oplus {\mathbb Z} w_2\oplus {\mathbb Z} w_3$ is a primitive sublattice of $\mathrm {E}_8(2)$ with diagonal Gram matrix such that $(w_1^2)=4a<0$ , $(w_2^2)=4c<0$ , $(w_3^2)=-4$ .

For $(1,1,1,0)$ , we take $(e+2af-ak,e+(2b-2a)f+(a-b)k+u,e+h,k)$ .

For $(1,0,1,1)$ , we take $(e+2af-ak,2bf-bk+w,e+h,e+k+h+w')$ , where ${\mathbb Z} w\oplus {\mathbb Z} w'$ is a primitive sublattice of $\mathrm {E}_8(2)$ such that $(w^2)=4c<0$ , $(w^{\prime 2})=-4$ , $(w.w')=0$ .

For $(1,1,1,1)$ , we take $(e+2af-ak,e+(2b-2a)f+(a-b)k+u,e+h,e+k+h+w')$ , where ${\mathbb Z} u\oplus {\mathbb Z} w'$ is a primitive sublattice of $\mathrm {E}_8(2)$ such that $(u^2)=4(a-b+c)<0$ , $(w^{\prime 2})=-4$ , $(u.w')=0$ .

$\rho =17$

Here, we have $T=\mathrm {U}(2)\oplus \mathrm {U}(2)\oplus (-4m)$ , where $m\geq 1$ . A standard basis is $\{x,y,x',y',t\}$ . Up to swapping the two copies of $\mathrm {U}(2)$ and swapping the elements of a standard basis of each $\mathrm {U}(2)$ it is enough to construct embeddings with the following labels:

$$ \begin{align*} (1,0,0,0,0), (1,1,0,0,0), (1,0,0,0,1), (1,1,0,0,1), (0,0,0,0,1), \end{align*} $$

$$ \begin{align*} (1,1,1,1,0), (1,1,1,1,1), (1,0,1,0,0), (1,1,1,0,0), (1,1,1,0,1), (1,0,1,0,1). \end{align*} $$

The first five embeddings are obtained as direct sums of a primitive embedding of $\mathrm {U}(2)\oplus (-4m)$ into $\mathrm {U}\oplus \mathrm {E}_8(2)$ and the identity embedding $\mathrm { U}(2)\tilde \longrightarrow \mathrm {U}(2)$ . The respective primitive embeddings of $\mathrm {U}(2)\oplus (-4m)$ into $\mathrm {U}\oplus \mathrm {E}_8(2)$ are given by sending $x,y,t$ to the following triples:

$$ \begin{align*} (e,2e+2f+u_1, v_1), (e,e+2f+u_2,v_2), (e,2e+2f+u_3,e+v_3), (e,e+2f+u_4,e+v_4). \end{align*} $$

Here, ${\mathbb Z} u_i\oplus {\mathbb Z} v_i$ is a primitive sublattice of $\mathrm {E}_8(2)$ such that:

$(u_1^2)=-8$ , $(v_1^2)=-4m$ , $(u_1.v_1)=0$ ;

$(u_2^2)=-4$ , $(v_2^2)=-4m$ , $(u_2.v_2)=0$ ;

$(u_3^2)=-8$ , $(v_3^2)=-4m$ , $(u_3.v_3)=-2$ ;

$(u_4^2)=-4$ , $(v_4^2)=-4m$ , $(u_4.v_4)=-2$ .

The embedding labeled $(0,0,0,0,1)$ can be obtained by sending $x,y,t$ to

$$ \begin{align*} (2e+2f+w_0,2e+2f+w_1,e+w_2), \end{align*} $$

where $w_0,w_1,w_2$ generate a primitive sublattice of $\mathrm {E}_8(2)$ with Gram matrix

$$ \begin{align*} \left(\begin{array}{ccc}-8&-6&-2\\-6&-8&-2\\-2&-2&-4m \end{array}\right). \end{align*} $$

Indeed, this matrix is negative-definite.

To construct the last six embeddings, we exhibit the images of $x,y,x',y',t$ . In the case of $(1,1,1,1,0)$ , we consider

$$ \begin{align*} (e,e+2f+k+w_0,e-h,e-h-k+w_1,w_2), \end{align*} $$

where $w_0,w_1,w_2$ generate a primitive sublattice of $\mathrm {E}_8(2)$ with diagonal Gram matrix such that $(w_0^2)=(w_1^2)=-4$ and $(w_2^2)=-4m$ .

In the case of $(1,1,1,1,1)$ , we take

$$ \begin{align*} (e,e+2f+k+w_0,e-h,e-h-k+w_1,e+w_2), \end{align*} $$

where $w_0,w_1,w_2$ generate a primitive sublattice of $\mathrm {E}_8(2)$ with the negative-definite Gram matrix

$$ \begin{align*} \left(\begin{array}{ccc}-4&0&-2\\0&-4&0\\-2&0&-4m \end{array}\right). \end{align*} $$

In the case of $(1,0,1,0,0)$ , we take $(e,2f+k,e-h,-k,w)$ , where w is a primitive element of $\mathrm {E}_8(2)$ with $(w^2)=-4m$ .

For $(1,1,1,0,0)$ , we take $(e,e+2f+k+u_2,e-h,-k,v_2)$ .

For $(1,1,1,0,1)$ , we take $(e,e+2f+k+u_4,e-h,-k,e+v_4)$ .

For $(1,0,1,0,1)$ , we take $(e,2e+2f+k+u_3,e-h,-k,e+v_3)$ .

Proof of Corollary B

(i) Let $E_1$ and $E_2$ be non-isogenous elliptic curves, and let $X=\mathrm {Kum}(E_1\times E_2)$ . By [O1, §4], we have $\mathrm {Aut}_{\mathrm {Hdg}}(T(X))=\{\pm 1\}$ and $|\mathcal Enr(X)|=15$ . (The 15 Enriques involutions can be described geometrically as the Lieberman involutions and the Kondo–Mukai involutions.) We have $\mathrm {{rk}}(T(X))=4$ , hence $|\mathrm {{Br}}(X)[2]\setminus \{0\}|=15$ .

(ii) Let C be a smooth projective curve of genus 2 such that $\mathrm {NS\,}(\mathrm {Jac}(C))\cong {\mathbb Z}$ . Let $X=\mathrm {Kum}(\mathrm {Jac}(C))$ . Condition $\mathrm {Aut}_{\mathrm {Hdg}}(T(X))=\{\pm 1\}$ is satisfied since the Picard rank of X is odd. Ohashi [O2] shows that $|\mathcal Enr(X)|=31$ and describes these 31 involutions geometrically. In this case $\mathrm {{rk}}(T(X))=5$ , so $|\mathrm {{Br}}(X)[2]\setminus \{0\}|=31$ .

Taking into account (i) and (ii), Corollary B follows from Theorem A since a surjective map of finite sets of the same cardinality is a bijection.

4 Singular K3 surfaces

K3 surfaces over $\overline {\mathbb Q}$

For a variety X over $\overline {\mathbb Q}$ and an element $g \in \mathrm {{Gal}}(\overline {\mathbb Q}/{\mathbb Q})$ , we define $X^g=X\times _{\overline {\mathbb Q}, g}\overline {\mathbb Q}$ . Then, we have a morphism $g\colon X\to X^g$ making the following diagram commutative:

Here, the vertical arrows are structure morphisms. A morphism of $\overline {\mathbb Q}$ -varieties $\phi \colon X\to Y$ gives rise to a morphism of $\overline {\mathbb Q}$ -varieties $\phi ^g=g\phi g^{-1}\colon X^g\to Y^g$ .

Let $K\subset \overline {\mathbb Q}$ be a subfield, and let $G_K=\mathrm {{Gal}}(\overline {\mathbb Q}/K)$ .

Definition 4.1. Let X be a variety over $\overline {\mathbb Q}$ .

1. (i) The field of moduli of X over K is the subfield $K(X)\subset \overline {\mathbb Q}$ fixed by the group $\{g\in G_K | X\cong X^g \}$ .

2. (ii) Let $B \subset \mathrm {{Br}}(X)$ be a finite subgroup. The field of moduli of the pair $(X, B)$ over K is the subfield $K(X,B)\subset \overline {\mathbb Q}$ fixed by the group

   $$ \begin{align*}\{g\in G_K| \exists \, \text{an isomorphism} \, f \colon X^g\rightarrow X \, \text{such that} \, (g^* \circ f^*)|_B= \mathrm{id}_B \}.\end{align*} $$

Let us fix an embedding $\overline {\mathbb Q}\subset {\mathbb C}$ . For a K3 surface X over $\overline {\mathbb Q}$ we write $T(X)$ for the transcendental lattice of $X_{\mathbb C}$ . One has natural isomorphisms ([CS, Prop. 5.2.3 and p. 142])

$$ \begin{align*}\mathrm{{Br}}(X) \cong \mathrm{{Br}}(X_{\mathbb C}) \cong \mathrm{{Hom}}(T(X), {\mathbb Q} / {\mathbb Z}). \end{align*} $$

Remark 4.2. Let X be a K3 surface over $\overline {\mathbb Q}$ of Picard rank at least $12$ . According to [V, Rem. 6.1(2), p. 32] a Hodge isometry $h \colon T(X^g) \tilde \rightarrow T(X)$ exists if and only if $X \cong X^g$ . It follows that in this case $K(X,B)$ is the fixed field of the group

$$ \begin{align*}\{g\in G_K | \exists \, \text{a Hodge isometry} \, h \colon T(X^g) \rightarrow T(X) \, \text{such that} \, (g^* \circ h^*)|_B = \mathrm{id}_B \}.\end{align*} $$

For a K3 surface over $\overline {\mathbb Q}$ , we have $\mathrm {Aut}(X)=\mathrm {Aut}(X_{\mathbb C})$ , since $Aut_{X/\overline {\mathbb Q}}$ is a discrete group scheme. Hence, the set of conjugacy classes of fixed point free involutions $\mathcal Enr(X)\subset \mathrm {Aut}(X)$ coincides with $\mathcal Enr(X_{\mathbb C})$ .

Proposition 4.3. Let X be a K3 surface over $\overline {\mathbb Q}$ such that $\mathrm {Aut}_{\mathrm {Hdg}}(T(X))=\{\pm 1\}$ . The Galois group $G_{K(X)}$ acts naturally on ${\mathcal Enr}(X)$ and on $\mathrm {{Br}}(X)[2]$ so that the map $\varphi _X\colon {\mathcal Enr}(X) \to \mathrm {{Br}}(X)[2]$ is $G_{K(X)}$ -equivariant.

Proof. Write $K := K(X)$ . We use $\sigma $ and $\tau $ to denote arbitrary elements of $G_K$ . By Definition 4.1(i), we can find an isomorphism $f_{\sigma ,\tau } \colon X^{\sigma } \tilde \longrightarrow X^{\tau }$ .

Let us denote the conjugacy class of $\psi \in \mathrm {Aut}(X)$ by $[\psi ]$ .

A fixed point free involution $\iota \colon X\to X$ gives rise to a fixed point free involution $\iota ^{\sigma }=\sigma \iota \sigma ^{-1}\colon X^{\sigma }\rightarrow X^{\sigma }$ , and one has $(\iota ^{\sigma })^{\tau }=\iota ^{\tau \sigma }$ . We define an action of $G_K$ on ${\mathcal Enr}(X)$ by making $\sigma $ send $[\iota ]$ to $[f_{1,\sigma }^{-1}\iota ^{\sigma } f_{1,\sigma }]$ . This class depends neither on the choice of $\iota $ in its conjugacy class, nor on the choice of $f_{1,\sigma }$ . We have

$$ \begin{align*} [f_{1,\tau}^{-1}(f_{1,\sigma}^{-1}\iota^{\sigma} f_{1,\sigma})^{\tau} f_{1,\tau}]= [(f_{1,\sigma}^{\tau} f_{1,\tau})^{-1}\iota^{\tau\sigma}(f_{1,\sigma}^{\tau} f_{1,\tau})] =[f_{1,\tau\sigma}^{-1}\iota^{\tau\sigma} f_{1,\tau\sigma}], \end{align*} $$

because $f_{1,\tau \sigma }$ and $f_{1,\sigma }^{\tau } f_{1,\tau }$ are both isomorphisms $X\tilde \longrightarrow X^{\tau \sigma }$ , so replacing one of them by the other does not change the conjugacy class.

Let us now define an action of $G_K$ on $\mathrm {{Br}}(X)[2]$ by making $\sigma \in G_{K}$ act as $f_{1,\sigma }^*(\sigma ^{-1})^*$ which is induced by $\sigma ^{-1}f_{1,\sigma }\colon X\to X^{\sigma }\to X$ . This action on $\mathrm {{Br}}(X)[2]$ does not depend on the choice of $f_{1,\sigma }$ . Indeed, $f_{1,\sigma }$ is well defined up to an automorphism of X, but the action of $\mathrm {Aut}(X)$ on $\mathrm {{Br}}(X)[2]$ factors through the action of $\mathrm {Aut}_{\mathrm { Hdg}}(T(X))$ . The latter group is $\{\pm 1\}$ by assumption, so $\mathrm {Aut}(X)$ acts on $\mathrm {{Br}}(X)[2]$ trivially. The map $(f_{1,\sigma })^{\tau }=\tau f_{1,\sigma }\tau ^{-1}$ is an isomorphism $X^{\tau }\tilde \longrightarrow X^{\tau \sigma }$ , hence $(f_{1,\sigma })^{\tau } f_{1,\tau }$ is an isomorphism $X\to X^{\tau \sigma }$ , so for the purpose of calculating the induced action of $\mathrm {{Br}}(X)[2]$ , we can replace it with $f_{1,\tau \sigma }$ . This shows that sending $\sigma \in G_{K}$ to the map induced on $\mathrm {{Br}}(X)[2]$ by $\sigma ^{-1}f_{1,\sigma }$ is indeed an action.

We have a commutative diagram

where the vertical maps are quotients by the respective fixed point free involutions. Thus the image of the nonzero element of $\mathrm {{Br}}(X/\iota )$ in $\mathrm {{Br}}(X)[2]$ followed by the action of $\sigma $ on $\mathrm {{Br}}(X)[2]$ is the same as the image of the nonzero element of $\mathrm {{Br}}(X/(f_{1,\sigma }^{-1}\iota ^{\sigma } f_{1,\sigma }))$ in $\mathrm {{Br}}(X)[2]$ . This proves that $\varphi _X$ is $G_{K}$ -equivariant.

Moduli fields of singular K3 surfaces

Let X be a singular K3 surface, that is, a K3 surface of maximal Picard rank $20$ . It is well known that every singular K3 surface is defined over $\overline {\mathbb Q}$ and has complex multiplication by the imaginary quadratic field $E = \mathrm {{End}}_{\mathrm {Hdg}} (T(X)_{\mathbb Q})$ . Assume that $\mathrm {{End}}_{\mathrm {Hdg}}(T(X))$ is the ring of integers $\mathcal {O}_E\subset E$ . In this situation, the results of [V] give explicit descriptions of the moduli fields $E(X)$ and $E(X, \mathrm {{Br}}(X)[n])$ which we now recall.

The group $\mathrm {{Br}}(X)\cong \mathrm {{Hom}}(T(X),{\mathbb Q}/{\mathbb Z})$ is naturally an $\mathcal {O}_E$ -module. Let $K_n/E$ be the ray class field of E with modulus $n\mathcal {O}_E,$ and let $\mathrm {Cl}_n(E) \cong \mathrm {Gal}(K_n/E)$ . The complex conjugation c acts on $\mathrm {Cl}_n(E)$ . Let $\mathrm {Cl}_n(E)^c$ be the c-invariant subgroup of $\mathrm {Cl}_n(E)$ . Define $\widetilde {K}_n \subset K_n$ as the fixed field of $\mathrm {Cl}_n(E)^c$ , so that $\mathrm {Gal}(\widetilde {K}_n/E) \cong \mathrm {Cl}_n(E)/ \mathrm {Cl}_n(E)^c$ . Note that $K_1$ is the Hilbert class field of E and $\mathrm {Cl}_1(E) = \mathrm {Cl}(E)$ is the usual class group. The complex conjugation c acts on $\mathrm {Cl}(E)$ as $-1$ .

Theorem 4.4. Let X be a singular K3 surface. Then $\widetilde {K}_n=E(X, \mathrm {{Br}}(X)[n])$ .

Proof. See [V, Th. 11.2 and Rem. 9.2 on p. 41].

In particular, we have $\widetilde {K}_1=E(X)$ . If n divides m, then $\widetilde {K}_n \subset \widetilde {K}_m$ .

Proof of Theorem C

The assumptions of Theorem C imply that $\mathrm {Aut}_{\mathrm {Hdg}}(T(X))={\mathcal O}_E^{\times }=\{\pm 1\}$ , so we can apply Proposition 4.3. Let $\rho $ be the representation of $G_{\widetilde {K}_1}$ in $\mathrm {{Br}}(X)[2] \cong ({\mathbb Z}/2)^2$ constructed in the proof of Proposition 4.3. It is enough to show that under our assumptions one has $|\rho (G_{\widetilde {K}_1})| = 3$ . Then $G_{\widetilde {K}_1}$ acts transitively on $\mathrm {{Br}}(X)[2]\setminus \{0\}$ , so in view of the $G_{\widetilde {K}_1}$ -equivariance established in Proposition 4.3 this will imply Theorem C. By Theorem 4.4, we need to prove that $[\widetilde K_2:\widetilde K_1]=3$ .

The following exact sequence describes the ray class group $\mathrm {Cl}_2(E)$ :

$$ \begin{align*} 0 \rightarrow \frac{\mathcal{O}_E^{\times}}{\{ x \in \mathcal{O}_E^{\times} | x \equiv 1 \, \mathrm{mod} \, 2 \}} \rightarrow (\mathcal{O}_E/2)^{\times} \rightarrow \mathrm{Cl}_2(E) \rightarrow \mathrm{Cl}(E) \rightarrow 0. \end{align*} $$

Under our assumptions, we have $\mathcal {O}_E^{\times } =\{ x \in \mathcal {O}_E^{\times } | x \equiv 1 \, \mathrm {mod} \, 2 \} = \{ \pm 1 \}$ . Since $2$ is inert in E, we have $\mathcal {O}_E / 2 \cong \mathbb {F}_4$ , and thus the sequence above becomes

$$ \begin{align*} 0 \rightarrow {\mathbb F}_4^{\times} \rightarrow \mathrm{Cl}_2(E) \rightarrow \mathrm{Cl}(E) \rightarrow 0. \end{align*} $$

This is a sequence of G-modules, where $G = \{1, c \}$ . We have $({\mathbb F}_4^{\times })^c =\{1\}$ and $\mathrm {H}^1(G, \mathbb {F}_4^{\times }) = 0$ , and hence $\mathrm {Cl}_2(E)^c = \mathrm {Cl}(E)^c$ . From this, we obtain the exact sequence

$$ \begin{align*} 0 \rightarrow {\mathbb F}_4^{\times} \rightarrow \mathrm{Gal}(\widetilde K_2/E) \rightarrow \mathrm{Gal}(\widetilde K_1/E) \rightarrow 0. \end{align*} $$

Thus, $[\widetilde K_2:\widetilde K_1]=3$ , as required.

Remark 4.5. When $2$ is split, a similar argument shows that the $G_{\widetilde {K}_1}$ -action on $\mathrm {{Br}}(X)[2]$ is trivial.

5 Constructing Enriques involutions

For a finite abelian group $G,$ we write $\ell (G)$ for the minimal number of generators of G. For a prime p we denote by $G_p$ the p-primary subgroup of G. Recall that for a lattice L we write $A_L=L^*/L$ for the discriminant group of L. When L is even, we denote by $q_L\colon A_L\to {\mathbb Q}/2{\mathbb Z}$ the finite quadratic form of L.

We need to recall fundamental results of Nikulin about the existence of lattices and their primitive embeddings.

Let $q \colon A \rightarrow {\mathbb Q} / 2 {\mathbb Z}$ be a finite quadratic form. The signature $\mathrm {sign}(q) \in {\mathbb Z} / 8 {\mathbb Z}$ of q is defined as $(t_+ - t_-) \bmod 8$ , where $(t_+, t_-)$ is the signature of any even lattice whose discriminant form is isomorphic to $(A,q)$ (such a lattice always exists and, moreover, this notion is well-defined). One also has

(1) $$ \begin{align} \mathrm{sign}(q \oplus q') = \mathrm{sign}(q) + \mathrm{sign}(q'). \end{align} $$

Write $A = \bigoplus _p A_p$ , where p ranges over the prime numbers. Then one has quadratic forms $q_p \colon A_p \rightarrow {\mathbb Q}_p / {\mathbb Z}_p$ when p is odd and $q_2 \colon A_2 \rightarrow {\mathbb Q}_2/ 2 {\mathbb Z}_2$ when $p=2$ . It is clear that q is the orthogonal direct sum of the forms $q_p$ .

For an odd prime p, a finite abelian p-group $A_p$ , and a quadratic form $q_p\colon A_p\to {\mathbb Q}_p/{\mathbb Z}_p$ , Nikulin [N, Th. 1.9.1] showed that there is a unique ${\mathbb Z}_p$ -lattice $K(q_p)$ of rank $\ell (A_p)$ whose quadratic form is isomorphic to $q_p$ .

When $p=2$ , the same result of Nikulin says the following. Let $q^{(2)}_{\theta } (2)$ be the discriminant quadratic form of the rank one ${\mathbb Z}_2$ -lattice $(2 \theta )$ , where $\theta \in {\mathbb Z}_2^{\times }$ . For a finite abelian 2-group $A_2$ and a quadratic form $q_2\colon A_2\to {\mathbb Q}_2/2{\mathbb Z}_2$ we have the following alternative. If $q_2$ splits as an orthogonal direct sum $q_2=q^{(2)}_{\theta } (2) \oplus q^{\prime }_2$ , then there are precisely two even ${\mathbb Z}_2$ -lattices of rank $\ell (A_2)$ whose quadratic form is isomorphic to $q_2$ . If such a splitting of $q_2$ does not exist, there is a unique ${\mathbb Z}_2$ -lattice $K(q_2)$ of rank $\ell (A_2)$ whose quadratic form is isomorphic to $q_2$ . The following result is [N, Th. 1.10.1].

Theorem 5.1 (Nikulin).

An even lattice with signature $(t_+, t_-)$ and quadratic form $q\colon A\to {\mathbb Q}/2{\mathbb Z}$ exists if and only if the following conditions are satisfied:

1. (1) $t_+ - t_- \equiv \mathrm {sign}(q) \mod 8$ ;

2. (2) $t_+, t_- \geq 0$ and $t_+ + t_- \geq \ell (A)$ ;

3. (3) $(-1)^{t_-} |A_p| \equiv \mathrm {discr} K(q_p) \bmod {\mathbb Z}_p^{\times 2}$ for the odd primes p such that $t_+ + t_- = \ell (A_p)$ ;

4. (4) $|A_2| \equiv \pm \mathrm {discr} K(q_2) \bmod {\mathbb Z}_2^{\times 2}$ if $t_+ + t_- = \ell (A_2)$ and $q_2 \neq q_{\theta }^{(2)}(2) \oplus q_2'$ for any $\theta $ and $q_2'$ .

The following result is a consequence of [N, Prop. 1.15.1] where we took into account that N is the unique lattice of signature $(2,10)$ whose quadratic form is isomorphic to $q_N$ (see [N, Cor. 1.13.4].

Theorem 5.2 (Nikulin).

Let L be an even lattice with signature $(2_+, k_-)$ and quadratic form $q_L\colon A_L\to {\mathbb Q}/2{\mathbb Z}$ . The existence of a primitive embedding $L \hookrightarrow N$ is equivalent to the existence of the following data:

• • subgroups $H_L \subset A_L$ and $H_N \subset A_N$ ;

• • an isomorphism of finite quadratic forms $\gamma \colon (H_L,q_L|_{H_L}) \xrightarrow {\sim } (H_N,q_N|_{H_N})$ ;

• • an even negative-definite lattice K of rank $10-k$ ;

• • an isomorphism of finite quadratic forms $\delta $ from $(A_K,-q_K)$ to the restriction of $q_L\oplus -q_N$ to $\Gamma _{\gamma }^{\perp } / \Gamma _{\gamma }$ , where the isotropic subgroup $\Gamma _{\gamma }\subset A_L\oplus A_N$ is the graph of $\gamma $ in $H_L\oplus H_N\subset A_L\oplus A_N$ .

Moreover, if $i \colon L \hookrightarrow N$ is a primitive embedding associated to $(H_L, H_N, \gamma, K, \delta )$ , then $K \cong i(L)^{\perp }$ .

Remark 5.3.

1. (1) If $f \colon \widetilde {K} \to K$ is an isomorphism of lattices and $\bar {f} \colon A_{\widetilde {K}} \to A_K$ is the induced isomorphism, then the primitive embeddings $L\hookrightarrow N$ associated to $(H_L, H_N, \gamma , K, \delta )$ and to $(H_L, H_N, \gamma , \widetilde {K},\delta \circ \bar {f})$ are isomorphic.

2. (2) An analog of Theorem 5.2 gives the conditions for the existence of a primitive embedding of $L\otimes {\mathbb Z}_p$ into $N\otimes {\mathbb Z}_p$ , for any prime p. The analog of (1) also holds in this context.

Definition 5.4. Let L be a lattice such that $0<\mathrm {{rk}}(L) \leq 10$ . We say that a sublattice $L' \subset L$ of finite index satisfies condition $(*)$ if

$$ \begin{align*} \mathrm{gcd}( 2\mathrm{discr}(L), [L \colon L']) = 1, \end{align*} $$

and for each prime p not dividing $ 2\mathrm {discr}(L),$ we have $\ell (A_{L',p}) < 12 - \mathrm {{rk}}(L')$ .

Proposition 5.5. Any lattice L such that $0<\mathrm {{rk}}(L) \leq 10$ contains infinitely many distinct sublattices $L'\subset L$ satisfying condition $(*)$ .

Proof. Let p be any odd prime not dividing $\mathrm {discr}(L)$ . As is well known (see, e.g., [N, Cor. 1.9.3]), the unimodular p-adic lattice $L\otimes {\mathbb Z}_p$ has an orthogonal ${\mathbb Z}_p$ -basis $v_1,\ldots ,v_n$ such that $(v_i^2)\in {\mathbb Z}_p^{\times }$ for $i=1,\ldots , n$ . The images of $v_1,\ldots ,v_n$ in $(L\otimes {\mathbb Z}_p)/p\cong L/p$ form a basis of this ${\mathbb F}_p$ -vector space. Let $L'\subset L$ be the inverse image of the hyperplane spanned by the images of $v_2,\ldots ,v_n$ . Thus $[L:L']=p$ , so that $\mathrm {discr}(L')=p^2\mathrm {discr}(L)$ . Since p does not divide $\mathrm { discr}(L)$ , we have a canonical isomorphism $A_{L'}\cong A_L\oplus A_{L',p}$ . It is enough to check that $\ell (A_{L',p})=1$ , which says that $A_{L',p}$ is cyclic. It is clear that $A_{L',p}\cong A_{L'\otimes {\mathbb Z}_p}$ , so it is enough to prove that $\mathrm {{Hom}}_{{\mathbb Z}_p}(L'\otimes {\mathbb Z}_p,{\mathbb Z}_p)/(L'\otimes {\mathbb Z}_p)\cong {\mathbb Z}/p^2$ . The ${\mathbb Z}_p$ -module $L'\otimes {\mathbb Z}_p$ is freely generated by $pv_1,v_2,\ldots ,v_n$ , hence the ${\mathbb Z}_p$ -module $\mathrm {{Hom}}_{{\mathbb Z}_p}(L'\otimes {\mathbb Z}_p,{\mathbb Z}_p)\subset L'\otimes {\mathbb Q}_p$ is freely generated by $p^{-1}v_1,v_2,\ldots ,v_n$ , which implies the result.

Condition $(*)$ implies that $[L \colon L']$ is odd, and hence the inclusion $L' \subset L$ induces a natural isomorphism

(2) $$ \begin{align} \mathrm{{Hom}}(L', {\mathbb Z} / 2 {\mathbb Z}) \cong \mathrm{{Hom}}(L, {\mathbb Z} / 2 {\mathbb Z}). \end{align} $$

Recall that for a primitive embedding $i\colon L\hookrightarrow N$ we denote by $i^*(\varepsilon )$ the precomposition of the character $\varepsilon \colon N\to {\mathbb Z}/2$ with i.

Theorem 5.6. Let $L' \subset L$ be an inclusion of even lattices of signature $(2_+, k_-)$ , where $0\leq k\leq 8$ . Then we have the following statements.

1. (a) If $L' \subset L$ satisfies condition $(*)$ , then for any primitive embedding $i\colon L\hookrightarrow N$ with $i^*(\varepsilon )\neq 0$ there exists a primitive embedding $i'\colon L'\hookrightarrow N$ such that $i^{\prime *}(\varepsilon )=i^*(\varepsilon )$ under the identification (2).

2. (b) If $[L \colon L']$ is odd, then for any primitive embedding $i'\colon L'\hookrightarrow N$ with $i^{\prime *}(\varepsilon )\neq 0$ there exists a primitive embedding $i\colon L\hookrightarrow N$ such that $i^{\prime *}(\varepsilon )=i^*(\varepsilon )$ under the identification (2).

Proof. (a) Let $i\colon L\hookrightarrow N$ be a primitive embedding such that $i^*(\varepsilon )\neq 0$ . Then $K:=i(L)^{\perp }_N$ is an even negative-definite lattice of rank $10-k$ . By Theorem 5.2, the embedding i corresponds to some datum $(H_L, H_N, \gamma , K, \delta )$ .

Since $L' \subset L$ satisfies condition $(*)$ , the index $[L:L']$ is coprime to $|A_L|$ , hence $A_{L'}$ canonically isomorphic to $A_L \oplus A_{\mathrm {new}}$ , where $|A_{\mathrm {new}}|=[L:L']^2$ . Then $q_{L'}$ is an orthogonal direct sum $q_{L'}\cong q_L\oplus q_{\mathrm {new}}$ , where $q_{\mathrm {new}}$ is a quadratic form on $A_{\mathrm {new}}$ .

We claim that there is a negative-definite lattice $K'$ of rank $10 - k$ such that $A_{K'}\cong A_K \oplus A_{\mathrm {new}}$ and $q_{K'}\cong q_K\oplus -q_{\mathrm {new}}$ . Since $L'$ is a sublattice of L of finite index and $\mathrm {{rk}}(K)=10-k$ , we have

$$ \begin{align*} \mathrm{sign}(q_L)\equiv \mathrm{sign}(q_{L'}) \bmod 8, \quad \quad k-10\equiv\mathrm{sign}(q_K) \bmod 8. \end{align*} $$

Since $q_{L'}\cong q_L \oplus q_{\mathrm {new}}$ , we have that $\mathrm {sign}(q_{L'}) = \mathrm {sign}(q_{L}) + \mathrm {sign}(q_{\mathrm {new}})$ by (1). Thus $\mathrm {sign}(q_{\mathrm {new}}) \equiv 0 \bmod 8$ , which implies property (1) of Theorem 5.1.

By condition $(*)$ , we know that $|A_{\mathrm {new}}|$ is odd and coprime to $|A_L|$ . For any odd prime p, the ${\mathbb Z}_p$ -lattices $L \otimes {\mathbb Z}_p$ and $K \otimes {\mathbb Z}_p$ are orthogonal complements of each other in the unimodular ${\mathbb Z}_p$ -lattice $N \otimes {\mathbb Z}_p$ , hence $|A_{L,p}|=|A_{K,p}|$ . Thus, $|A_K|$ and $|A_{\mathrm {new}}|$ are coprime. This implies

$$ \begin{align*} \ell(A_K \oplus A_{\mathrm{new}}) = \max\{\ell(A_K), \ell(A_{\mathrm{new}})\}\leq 10-k, \end{align*} $$

since $\ell (A_K)\leq \mathrm {{rk}}(K)=10-k$ and $\ell (A_{\mathrm {new}})\leq \ell (A_{L'})< 12-\mathrm {{rk}}(L)$ by condition $(*)$ . Thus, property (2) of Theorem 5.1 also holds.

We now check properties (3) and (4) taking into account the coprimality of $|A_K|$ and $|A_{\mathrm {new}}|$ . If p divides $|A_K|$ , then (3) and (4) hold because they hold for $A_K$ . If p divides $|A_{\mathrm {new}}|$ , then $\ell ( A_{\mathrm {new}}) < \mathrm {{rk}}(K')$ by condition $(*)$ , so there is nothing to check.

Theorem 5.1 now implies the existence of $K'$ with required properties.

Let us construct a datum defining the desired primitive embedding $L' \hookrightarrow N$ . Since $2A_N=0$ , we have $2H_N=0$ and thus $2H_L=0$ , so that $H_L \subset A_{L,2}$ . In view of the canonical isomorphism $A_{L,2} \cong A_{L',2}$ , we can keep the same $H_{L'}=H_L$ , $H_N$ and $\gamma '=\gamma $ as the first three entries of our datum.

Recall that $A_{L'}\cong A_L \oplus A_{\mathrm {new}}$ . We have

$$ \begin{align*}\Gamma_{\gamma'} = \Gamma_{\gamma} \oplus 0 \subset \Gamma_{\gamma'}^{\perp} = \Gamma_{\gamma}^{\perp} \oplus A_{\mathrm{new}} \subset (A_L \oplus A_N) \oplus A_{\mathrm{new}},\end{align*} $$

hence $\Gamma _{\gamma '}^{\perp } / \Gamma _{\gamma '} = \Gamma _{\gamma }^{\perp } / \Gamma _{\gamma }\oplus A_{\mathrm {new}} \cong A_K \oplus A_{\mathrm {new}}$ . The restriction of

$$ \begin{align*} q_{L'}\oplus -q_N\cong(q_L\oplus -q_N)\oplus q_{\mathrm{ new}} \end{align*} $$

to $\Gamma _{\gamma '}^{\perp } / \Gamma _{\gamma '}$ is isomorphic to $-q_K\oplus q_{\mathrm {new}}$ via the isomorphism $\delta ':=(\delta , \mathrm {id})$ .

Take a negative-definite lattice $K'$ of rank $10 - k$ as above, that is, such that $A_{K'}\cong A_K \oplus A_{\mathrm {new}}$ and $q_{K'}\cong q_K\oplus -q_{\mathrm {new}}$ . Let $i'\colon L'\hookrightarrow N$ be a primitive embedding associated to the datum $(H_{L'}, H_{N}, \gamma ', K', \delta ')$ .

To prove that $i^{\prime *}(\varepsilon )=i(\varepsilon )$ under the natural identification (2), it is enough to show that the induced embeddings of ${\mathbb Z}_2$ -lattices $i_2\colon L\otimes {\mathbb Z}_2\hookrightarrow N\otimes {\mathbb Z}_2$ and $i^{\prime }_2\colon L'\otimes {\mathbb Z}_2\hookrightarrow N\otimes {\mathbb Z}_2$ are isomorphic.

First, we claim that $K \otimes {\mathbb Z}_2$ and $K' \otimes {\mathbb Z}_2$ are isomorphic ${\mathbb Z}_2$ -lattices. Since K and $K'$ are negative-definite of the same rank, and $|A_{K'}| = |A_K| \cdot |A_{\mathrm {new}}|$ , we have $\mathrm {discr}(K')= \mathrm {discr}(K)\cdot |A_{\mathrm {new}}|$ . Since $|A_{\mathrm {new}}|$ is a square of an odd integer, the even $2$ -adic lattices $K \otimes {\mathbb Z}_2$ and $K' \otimes {\mathbb Z}_2$ have the same rank, the same discriminant form, and the same discriminant modulo ${\mathbb Z}_2^{\times 2}$ . This implies that the ${\mathbb Z}_2$ -lattices $K \otimes {\mathbb Z}_2$ and $K' \otimes {\mathbb Z}_2$ are isomorphic (see [Nik79, Cor. 1.9.3]).

It remains to show that after tensoring with ${\mathbb Z}_2$ the data $(H_{L}, H_{N}, \gamma , K, \delta )$ and $(H_{L'}, H_{N}, \gamma ', K', \delta ')$ give rise to isomorphic embeddings of $L' \otimes {\mathbb Z}_2 \cong L \otimes {\mathbb Z}_2$ into $N\otimes {\mathbb Z}_2$ . The first three entries of each datum are the same. By Remark 5.3, it is enough to find an isomorphism of ${\mathbb Z}_2$ -lattices $f \colon K' \otimes {\mathbb Z}_2 \rightarrow K \otimes {\mathbb Z}_2$ such that $\delta ^{\prime }_2 =\delta _2 \circ \bar {f}$ . The existence of such an f follows from [N, Th. 1.9.5]. This concludes the proof of (a).

(b) Write $A := A_L = A_2 \oplus A_{\mathrm {odd}}$ , where $A_2$ is the 2-primary subgroup of A. Similarly, write $A' := A_{L'} = A^{\prime }_2 \oplus A^{\prime }_{\mathrm {odd}}$ . It is clear that $A_2\cong A^{\prime }_2$ . Then $q_{L'}$ is an orthogonal direct sum of quadratic forms $q_{L,2}$ on $A_2$ and $q_{\mathrm {odd}}$ on $A^{\prime }_{\mathrm {odd}}$ .

The overlattice L of $L'$ defines an isotropic subgroup $I \subset A'$ , where $|I|=[L:L']$ , so that $q_L$ is the quadratic form induced by $q_{L'}$ on $A= I^{\perp }/I$ . Since $[L \colon L']$ is odd by assumption, we have $I \subset A^{\prime }_{\mathrm {odd}}$ . Thus $I^{\perp } = A_2 \oplus I^{\perp }_{\mathrm {odd}}$ , where $I^{\perp }_{\mathrm {odd}}=I^{\perp }\cap A^{\prime }_{\mathrm {odd}}$ . This shows that $A= A_2 \oplus (I^{\perp }_{\mathrm {odd}}/I)$ .

Let $i'\colon L'\hookrightarrow N$ be a primitive embedding such that $i^{\prime *}(\varepsilon )\neq 0$ . Then $K':=i(L')^{\perp }_N$ is an even negative-definite lattice of rank $10-k$ . Let $(H_{L'}, H_N, \gamma ', K', \delta ')$ be a datum associated to $i'\colon L'\hookrightarrow N$ as in Theorem 5.2. In particular, $\delta '$ is an isomorphism of $-q_{K'}$ with the restriction of $q_{L'}\oplus -q_N$ to $\Gamma _{\gamma '}^{\perp } / \Gamma _{\gamma '}$ . Since $2A_N=0$ , we have $2H_{L'}=0$ , so that $H_{L'} \subset A^{\prime }_2=A_2$ . Hence $\Gamma _{\gamma '} \subset A_2\oplus A_N \subset A' \oplus A_N$ and thus $\Gamma _{\gamma '}^{\perp } = (\Gamma _{\gamma '}^{\perp })_2 \oplus A^{\prime }_{\mathrm {odd}}$ , where $(\Gamma _{\gamma '}^{\perp })_2=\Gamma _{\gamma '}^{\perp }\cap (A_2 \oplus A_N)$ . This shows that $\delta '$ identifies the finite quadratic form $-q_{K'}$ on $A_{K'}$ with the restriction of $(q_{L,2}\oplus -q_N)\oplus q_{\mathrm {odd}}$ to $\big ((\Gamma _{\gamma '}^{\perp }\big )_2/\Gamma _{\gamma '}) \oplus A^{\prime }_{\mathrm {odd}}$ .

The isotropic subgroup $I \subset A^{\prime }_{\mathrm {odd}}$ gives rise, via $\delta '$ , to an isotropic subgroup in $A_{K'}$ . The latter defines an overlattice $K' \subset K$ with $[K:K']=[L:L']$ , so that $\delta '$ induces an isomorphism $\delta $ of the quadratic form $-q_K$ on $A_K$ with the restriction of $(q_{L,2}\oplus -q_N)\oplus q_{\mathrm {odd}}$ to $\big ( (\Gamma _{\gamma '}^{\perp })_2/\Gamma _{\gamma '}\big )\oplus (I^{\perp }_{\mathrm {odd}}/I)$ . Let $i\colon L\hookrightarrow N$ be a primitive embedding associated to the datum $(H_{L}, H_N, \gamma , K, \delta )$ , where $H_L=H_{L'}$ and $\gamma =\gamma '$ .

To complete the proof of (b), it remains to show that i and $i'$ induce isomorphic embeddings of ${\mathbb Z}_2$ -lattices. This is proved by the same arguments as in (a).

Corollary 5.7. Let L be an even lattice of signature $(2_+, k_-)$ , where $0\leq k\leq 8$ . Write ${\mathcal S}(L)$ for the set of nonzero homomorphisms $\alpha \colon L\to {\mathbb Z}/2$ such that there is a primitive embedding $i\colon L\hookrightarrow N$ with $\alpha =i^*(\varepsilon )$ . Let $L'$ be a sublattice of L that satisfies condition $(*)$ . Then, under the natural identification $\mathrm {{Hom}}(L,{\mathbb Z}/2) \cong \mathrm {{Hom}}(L', {\mathbb Z}/2)$ , we have ${\mathcal S}(L)={\mathcal S}(L')$ .

Proof. Part (a) of Theorem 5.6 implies ${\mathcal S}(L)\subset {\mathcal S}(L')$ , whereas part (b) implies ${\mathcal S}(L')\subset {\mathcal S}(L)$ since $[L:L']$ is odd.

Proof of Theorem D

By Proposition 5.5, there are infinitely many sublattices $T \subset T(X)$ with pairwise different discriminants that satisfy condition $(*)$ . Endow T with the Hodge structure coming from $T(X)$ . Since $\mathrm {{rk}}(T) \leq 10$ , by [N, Th. 1.14.4], there exists a unique primitive embedding of the lattice T into the K3 lattice $\Lambda $ . We equip $\Lambda $ with the Hodge structure induced by the Hodge structure on T so that $T^{\perp }_{\Lambda }\subset \Lambda ^{(1,1)}$ . By the surjectivity of the period map, there is a K3 surface Y together with a Hodge isometry between $\Lambda $ and $\mathrm {H}^2(Y,{\mathbb Z})$ . The transcendental lattice $T(Y)$ is the orthogonal complement to $\mathrm {H}^2(Y,{\mathbb Z})^{(1,1)}$ , hence $T(Y)\cong T$ .

Applying Corollary 5.7 with $L=T(X)$ , we obtain ${\mathcal S}(T(X))={\mathcal S}(T(Y))$ . Now Corollary 2.6(i) (whose proof uses Lemma 2.3) gives $\mathrm {{Im}}(\Phi _X)\setminus \{0\}=\mathrm {{Im}}(\Phi _Y)\setminus \{0\}$ .