2.1 CM pairs

A number field E is called a CM field if it is an imaginary quadratic extension of a totally real number field. A CM algebra is a finite product of CM fields. A CM type on a CM algebra E is a subset $\Phi =\{\phi _1,\dots ,\phi _g\} \subseteq \text {Hom}(E,{\mathbb C})$ such that $\text {Hom}(E,{\mathbb C})=\Phi \sqcup \overline {\Phi }$ (where $\overline {\Phi } = \{\bar {\phi }_1,\ldots ,\bar {\phi }_g\}$ is the complex conjugate of $\Phi $ ).

A CM pair is a pair $(E,\Phi )$ consisting of a CM algebra E and a CM type $\Phi $ . Two CM pairs $(E,\Phi )$ and $(E',\Phi ')$ are said to be isomorphic if there exists an isomorphism $\alpha \colon E \rightarrow E'$ of ${\mathbb Q}$ -algebras such that $\phi \circ \alpha \in \Phi $ whenever $\phi \in \Phi '$ . To each CM pair $(E,\Phi )$ , we can associate a CM abelian variety $A_{(E,\Phi )} := {\mathbb C}^{[E:{\mathbb Q}]/2}/\Phi ({\mathcal O}_E)$ with ${\mathcal O}_E$ the ring of integers of E. It is known that $A_{(E,\Phi )}$ is defined over $\overline {\mathbb {Q}}$ .

A CM pair $(E,\Phi )$ is said to be primitive if there does not exist a proper sub-CM algebra $E_0$ of E such that $\Phi |_{E_0} := \{\phi |_{E_0} : \phi \in \Phi \}$ is a CM type on $E_0$ . A CM pair $(E,\Phi )$ is primitive if and only if $A_{(E,\Phi )}$ is simple.

Finally, the Galois closure $E^c$ of a CM algebra E is defined as follows: $E = E_1^{n_1}\times \cdots \times E_m^{n_m}$ with each $E_j$ a CM field, and $E^c$ is defined to be the composite of the Galois closures of $E_1,\ldots ,E_m$ in $\overline {\mathbb {Q}}$ . Then the Galois group $G := \mathrm {Gal}(E^c/{\mathbb Q})$ acts on $\text {Hom}(E,{\mathbb C}) = \Phi \sqcup \overline {\Phi }$ .

2.2 CM pair associated with A

We say that A is associated with a CM pair $(E,\Phi )$ , if A is isogeneous to $A_{(E,\Phi )} = {\mathbb C}^{|\Phi |}/\Phi ({\mathcal O}_E)$ with ${\mathcal O}_E$ the ring of integers of E.

Assume A is simple, then $E:= \mathrm {End}(A)\otimes {\mathbb Q}$ is a CM field of degree $2g$ . It is well-known that there exists a CM type $\Phi $ on E such that A is associated with the CM pair $(E,\Phi )$ , this CM pair $(E,\Phi )$ is primitive, and each primitive CM pair is obtained in this way.

In general, A is isogenous to $A_1^{n_1}\times \cdots \times A_m^{n_m}$ for some pairwise non-isogenous simple abelian varieties $A_1, \ldots , A_m$ . Each $A_j$ is associated with some CM pair $(E_j, \Phi _j)$ with $E_j$ a CM field. Set $E := E_1^{n_1} \times \cdots \times E_m^{n_m}$ , with $E_j^{(k)}$ the k-th component of $E_j^{n_j}$ and $p_j^{(k)}$ the natural projection $E \rightarrow E_j^{(k)}$ , and

(2.1) $$ \begin{align} \Phi := \bigsqcup_{j=1}^m \Phi_j^{\sqcup n_j}, \quad \text{ where } \Phi_j^{\sqcup n_j} := \{\Phi_j \circ p_j^{(k)} : k \in \{1,\ldots,n_j\}\}. \end{align} $$

Then $(E,\Phi )$ is a CM pair and A is isogeneous to $A_{(E,\Phi )} = {\mathbb C}^g/\Phi ({\mathcal O}_E)$ with ${\mathcal O}_E = {\mathcal O}_{E_1}^{n_1}\times \cdots \times {\mathcal O}_{E_m}^{n_m}$ . So A is associated with $(E,\Phi )$ .

2.3 Algebraic tori associated with A

Use the notation from last subsection. In this subsection, we associate with A three algebraic tori $\mathrm {MT}(A) \subseteq T \subseteq E^\times $ defined over ${\mathbb Q}$ .

Let $V = H^1(A,{\mathbb Q})$ . It is a ${\mathbb Q}$ -vector space of dimension $2g$ with a polarization $\psi $ which is a non-degenerate anti-symmetric pairing $V \times V \rightarrow {\mathbb Q}$ .

The first algebraic torus is $E^\times := (E_1^\times )^{n_1} \times \cdots \times (E_m^\times )^{n_m}$ , where $E_j^\times := \mathrm {Res}_{E_j/{\mathbb Q}}{\mathbb G}_{\mathrm {m},E_j }$ . Assume $m=1$ and $n_1 = 1$ , i.e. A is simple. Then the action of $E= E_1$ on V makes V into a one-dimensional E-vector space. Therefore $E^\times $ acts on V and every character of $E^\times $ occurs with multiplicity $1$ in $V_{{\mathbb C}}$ . In this case, the group of characters of $E^\times $ is $X^*(E^\times )={\mathbb Z}[\text {Hom}(E,{\mathbb C})] = \oplus _{\phi \in \Phi } {\mathbb Z} \phi \bigoplus \oplus _{\phi \in \Phi } {\mathbb Z} \overline {\phi }$ . For arbitrary A, we still have

$$\begin{align*}X^*(E^\times) = \oplus_{\phi \in \Phi} {\mathbb Z} \phi \bigoplus \oplus_{\phi\in \Phi} {\mathbb Z} \overline{\phi}. \end{align*}$$

Thus $E^\times $ can be identified with the diagonal torus of $\mathrm {GL}(V_{{\mathbb C}})$ , under

(2.2) $$ \begin{align} V_{{\mathbb C}} = \oplus_{\phi \in \Phi} V_{{\mathbb C},\phi} \bigoplus \oplus_{\phi\in \Phi} V_{{\mathbb C},\overline{\phi}}. \end{align} $$

The second algebraic torus T is defined as $E^\times \bigcap \mathrm {GSp}(V,\psi )$ , and is a maximal torus of $\mathrm {GSp}(V,\psi )$ and hence has dimension $g+1$ . The restriction of the morphism $\lambda \colon \mathrm {GSp}_{2g} \rightarrow \mathbb {G}_{\mathrm {m}}$ , $h \mapsto \det (h)^{1/g}$ , to T is a character $\epsilon _0 \in X^*(T) = \text {Hom}(T,\mathbb {G}_{\mathrm {m}})$ . By abuse of notation we write $\phi $ for $\phi |_T$ for each $\phi \in \Phi $ . The character group of T is

(2.3) $$ \begin{align} X^*(T) = {\mathbb Z}\epsilon_0\bigoplus \oplus_{\phi \in \Phi} {\mathbb Z} \phi. \end{align} $$

Note that T splits over the Galois closure $E^c$ of E, i.e. $T_{E^c} \simeq \mathbb {G}_{\mathrm {m}, E^c}^{g+1}$ . Hence the Galois group $\mathrm {Gal}(E^c/{\mathbb Q})$ acts on $X^*(T)$ .

Before moving on, let us point out that the cocharacter group of $(E_1^\times )^{n_1} \times \cdots \times (E_m^\times )^{n_m}$ , resp. of T, can be written in the same way as the character group, and hence contains the element $\epsilon _0$ .

Finally, let us turn to the Mumford–Tate group of A denoted by $\mathrm {MT}(A)$ , i.e. the smallest ${\mathbb Q}$ -subgroup of $\mathrm {GL}(V)$ such that the map $\alpha \colon {\mathbb S}\longrightarrow \mathrm {GL}(V)_{{\mathbb R}}$ defining the Hodge structure on V factors through $\mathrm {MT}(A)_{{\mathbb R}}$ . It is known that $\alpha ({\mathbb S}) \subseteq ((E_1^\times )^{n_1} \times \cdots \times (E_m^\times )^{n_m})_{{\mathbb R}}$ and $\alpha ({\mathbb S})\subseteq \mathrm {GSp}(V,\psi )_{{\mathbb R}}$ . Hence $\mathrm {MT}(A) \subseteq T$ .

2.4 Reflex

Let $(E,\Phi )$ be a CM pair, and let $E^c$ be the Galois closure of E (see the end of §2.1 for definition). Then for each $\sigma \in G:=\mathrm {Gal}(E^c/{\mathbb Q})$ , the set $\sigma \Phi := \{\sigma \circ \phi : \phi \in \Phi \}$ is still a CM type on E. Thus we can define the stabilizer $\mathrm {Stab}_G(\Phi )$ of $\Phi $

The reflex field $E^*$ of $(E,\Phi )$ is defined to be $(E^c)^{\mathrm {Stab}_G(\Phi )}$ . One can also characterize $E^*$ as the subfield of $E^c$ generated by the elements $\sum _{\phi \in \Phi } \phi (a)$ with all $a \in E$ . Note that $\mathrm {Gal}(E^c/E^*) = \mathrm {Stab}_G(\Phi )$ .

The reflex CM type on $E^*$ is defined as follows. We have $\text {Hom}(E^c, {\mathbb C}) = \text {Hom}(E^c, E^c) = G$ . Thus $\Phi _{E^c} := \{\sigma \in G : \sigma |_E \in \Phi \} \subseteq G$ is a CM type on $E^c$ . Hence under $\text {Hom}(E^*, {\mathbb C}) \simeq G/\mathrm {Stab}_G(\Phi )$ , the set $\Phi ^* := \{\sigma ^{-1} \mathrm {Stab}_G(\Phi ) : \sigma \in G \text { with } \sigma |_E \in \Phi \}$ is the reflex CM type on  $E^*$ .

We call $(E^*,\Phi ^*)$ the reflex CM pair of $(E,\Phi )$ . Note that $E^*$ is a CM field, in contrast to E. Better, $(E^*,\Phi ^*)$ is a primitive CM pair.

The reflex norm is the morphism

(2.4) $$ \begin{align} \mathrm{rec} \colon (E^*)^{\times} \rightarrow E^\times, \quad a \mapsto \prod\nolimits_{\psi \in \Phi^*} \psi(a). \end{align} $$

Notice that this morphism can be viewed either as a homomorphism of groups or a morphism of algebraic tori defined over ${\mathbb Q}$ .

Now assume $(E,\Phi )$ is a primitive CM pair. Then $(E^{**}, \Phi ^{**}) = (E,\Phi )$ . Moreover, $\mathrm {MT}(A_{(E,\Phi )}) \simeq \mathrm {MT}(A_{(E^*,\Phi ^*)})$ , and the image of $\mathrm {rec}$ is $\mathrm {MT}(A_{(E,\Phi )})$ .

2.5 Periods of CM abelian varieties

Let A be a CM abelian variety of dimension g, associated with the CM pair $(E,\Phi )$ as constructed in §2.2. The de Rham cohomology $H_{\mathrm {dR}}^1(A)$ is a $\overline {\mathbb {Q}}$ -vector space since A is defined over $\overline {\mathbb {Q}}$ , and it has an E-eigenbasis $\{\omega _1,\ldots ,\omega _g,\eta _1,\ldots ,\eta _g\}$ . As in (2.2), the Betti cohomology $H^1(A,\overline {\mathbb {Q}}) = H^1(A,{\mathbb Q})\otimes \overline {\mathbb {Q}}$ has an E-eigenbasis $\{\phi _1,\ldots ,\phi _g,\overline {\phi }_1,\ldots ,\overline {\phi }_g\}$ .

The period torsor $\mathcal {B}_A$ is an $\mathrm {MT}(A)$ -torsor representing $ \underline {\mathrm {Iso}}^{\otimes }(H_{\mathrm {dR}}^1(A), H^1(A,\overline {\mathbb {Q}})) $ . The de Rham–Betti comparison isomorphism

$$ \begin{align*}\beta \colon H_{\mathrm{dR}}^1(A_{{\mathbb C}})\longrightarrow H^1(A,{\mathbb C}) \end{align*} $$

is a complex point of $\mathcal {B}_A$ . We computed in [Reference Gao, Ullmo and YafaevGUY24, Rmk. 6.5] that, up to scalars in $\overline {\mathbb {Q}}^\times $ , $\beta $ is diagonalized under the basis $\{\omega _1,\ldots ,\omega _g,\eta _1,\ldots ,\eta _g\}$ of $H_{\mathrm {dR}}^1(A)$ and the basis $\{\phi _1,\ldots ,\phi _g, \overline {\phi }_1,\ldots ,\overline {\phi }_g\}$ of $H^1(A,\overline {\mathbb {Q}})$ with coordinates

(2.5) $$ \begin{align} \left(\theta_1,\dots, \theta_g, \frac{2\pi i}{\theta_1}, \dots, \frac{2\pi i}{\theta_g} \right). \end{align} $$

The map $(\sum _j a_j\omega _j + \sum _l b_l\eta _l \mapsto \sum _j a_j \phi _j + \sum _l b_l \overline {\phi }_l)$ is a $\overline {\mathbb {Q}}$ -point of $\mathcal {B}_A$ . Using this as a base point, we get an isomorphism $ \mathcal {B}_A \xrightarrow {\sim } \mathrm {MT}(A)_{\overline {\mathbb {Q}}}$ which sends $\beta $ to the diagonal matrix (2.5). By abuse of notation, denote this diagonal matrix still by $\beta $ . Then $\beta \in \mathrm {MT}(A)({\mathbb C})$ , and we have

(2.6) $$ \begin{align} \phi_j(\beta) =\theta_j ~~(\forall j), \quad \overline{\phi}_j(\beta) = 2\pi i/\theta_j ~~(\forall j), \quad \epsilon_0(\beta) = 2\pi i \end{align} $$

with all $\phi _j$ , $\overline {\phi }_j$ viewed as characters of $E^\times $ (or of T) and $\epsilon _0 \in X^*(T)$ .

The following lemma is an immediate consequence of the discussion above. Consider the exact sequences obtained from $\mathrm {MT}(A) < T$ and $\mathrm {MT}(A) < E^\times $ respectively

(2.7) $$ \begin{align} 0\longrightarrow N\longrightarrow X^*(T)\longrightarrow X^*(\mathrm{MT}(A))\longrightarrow 0 \\ 0\longrightarrow N'\longrightarrow X^*(E^\times)\longrightarrow X^*(\mathrm{MT}(A))\longrightarrow 0. \nonumber \end{align} $$

Any non-zero vector in N (resp. in $N'$ ) produces a non-trivial monomial relation between the holomorphic periods of A and $\pi $ (resp. between the holomorphic and the anti-holomorphic periods of A and $\pi $ ). More precisely, let $M_{\mathrm {hol}}$ be the multiplicative group of monomials in the ideal of algebraic relations between $\theta _1,\ldots ,\theta _g,\pi $ , and let $M_{\mathrm {full}}$ be the multiplicative group of monomials in the ideal of algebraic relations between $\theta _1,\ldots ,\theta _g,\frac {2\pi i}{\theta _1},\ldots ,\frac {2\pi i}{\theta _g}, \pi $ .

In particular, if $T=\mathrm {MT}(A)$ no relations among the holomorphic periods are obtained from N, and the set of relations among the holomorphic and anti-holomorphic periods are generated by $\theta _j \frac {2\pi i}{\theta _j}\simeq 2\pi i$ for $j \in \{1,\ldots ,g\}$ .

In the case where A is simple, we can do better.

Lemma 2.3. Assume that A is a simple CM abelian variety. Then the map $X^*(E^\times ) \rightarrow X^*(\mathrm {MT}(A))$ factors through $\mathrm {rec}^*$ obtained from the reflex norm (2.4), and $N' = \ker (\mathrm {rec}^*)$ . In other words, we have the short exact sequence

$$\begin{align*}0 \rightarrow N' \rightarrow X^*(E^\times) \xrightarrow{\mathrm{rec}^*} X^*((E^*)^\times). \end{align*}$$

3.1 Hodge rings of CM abelian varieties

Let $B^p(A):=H^{p,p}(A,{\mathbb C})\cap H^{2p}(A,{\mathbb Q})$ be the ${\mathbb Q}$ -vector space of ${\mathbb Q}$ -Hodge cycles of type $(p,p)$ on A. The Hodge ring of A is defined to be

(3.1) $$ \begin{align} B(A) := \bigoplus_{p=0}^{g}B^p(A). \end{align} $$

Let us recall the following description of $B(A)$ by Pohlmann’s Theorem [Reference Pohlmann.Poh68]. Let $S=\Phi \sqcup \overline {\Phi }$ . We have an isomorphism $H^r(A,{\mathbb C})=\bigwedge ^r {\mathbb C}^S$ and

$$ \begin{align*}H^{1,0}(A,{\mathbb C})= \sum_{\phi \in \Phi} {\mathbb C} \phi \text{ and } H^{0,1}(A,{\mathbb C})= \sum_{\phi' \in \overline{\Phi}} {\mathbb C} \phi'. \end{align*} $$

Fix an ordering on S such that $\phi \le \phi '$ for each $\phi \in \Phi $ and $\phi ' \in \overline {\Phi }$ . Let ${\mathcal P}(S) $ be the set of subsets of S. For each subset $P\in {\mathcal P}(S)$ , let $[P]:=\bigwedge _{\phi \in P}\phi .$ Then the component

$$ \begin{align*}H^{p,q}(A,{\mathbb C})= \bigwedge^p H^{1,0}(A,{\mathbb C})\otimes \bigwedge^q H^{0,1}(A,{\mathbb C}) \end{align*} $$

of the Hodge decomposition has a basis consisting of the $[P]$ such that $\vert P\cap \Phi \vert =p$ and $\vert P\cap \overline {\Phi }\vert =q$ for P an ordered subset of S. Finally, the Galois group $G=\mathrm {Gal}(E^c/{\mathbb Q})$ operates on ${\mathcal P}(S)$ .

Theorem 3.1 (Pohlmann).

For each $p \ge 0$ , the vector space $B^p(A)\otimes {\mathbb C}$ has a basis consisting of $[P]$ for those ordered sets $P\in {\mathcal P}(S)$ with $|P| = 2p$ such that

(3.2) $$ \begin{align} \vert \sigma P\cap \Phi\vert=\vert \sigma P\cap \overline{\Phi}\vert \quad \text{ for all } \sigma \in G. \end{align} $$

In particular $\dim _{{\mathbb Q}}B^p(A)$ is the number of ordered $P\in {\mathcal P}(S)$ with $\vert P\vert =2p$ satisfying (3.2).

3.2 Kernel of the restriction of character groups

Consider the short exact sequence

(3.3) $$ \begin{align} 0 \rightarrow N \rightarrow X^*(T) \rightarrow X^*(\mathrm{MT}(A)) \rightarrow 0. \end{align} $$

Each element $\alpha \in X^*(T)$ can be written in a unique way as $\sum _{\phi \in \Phi } a_{\phi } \phi +a\epsilon _0$ with $a_{\phi } \in {\mathbb Z}$ , and we denote by $n(\alpha ) := \max \{|a_{\phi }|\}$ .

For each $n \ge 1$ , define

$$\begin{align*}X^*(T)_n := \{\alpha \in X^*(T) : n(\alpha) \le n \}. \end{align*}$$

Using the notation from $\S $ 2.2, the CM abelian variety $A^n$ has CM type $\Phi ^{\sqcup n}$ . For each $\ell \in \{1,\ldots ,n\}$ , denote by $\Phi ^{(\ell )}$ the $\ell $ -th component $\Phi $ in $\Phi ^{\sqcup n}$ , and for each $\phi \in \Phi $ denote by $\phi ^{(\ell )}$ the element $\phi $ in $\Phi ^{(\ell )}$ . Fix an ordering on $\Phi \cup \overline {\Phi }$ such that $\phi \le \phi '$ for each $\phi \in \Phi $ and $\phi ' \in \overline {\Phi }$ . It induces an ordering on $\Phi ^{\sqcup n} \cup \overline {\Phi }^{\sqcup n}$ as follows:

• • On each $\Phi ^{(\ell )} \simeq \Phi $ (resp. $\overline {\Phi }^{(\ell )}\simeq \overline {\Phi }$ ) it is the one on $\Phi $ (resp. on $\overline {\Phi }$ );

• • For $\ell \le \ell '$ , we have $\phi \le \phi '$ for all $\phi \in \Phi ^{(\ell )}$ and $\phi \in \Phi ^{(\ell ')}$ ;

• • For $\ell \le \ell '$ , we have $\phi \le \phi '$ for all $\phi \in \overline {\Phi }^{(\ell )}$ and $\phi \in \overline {\Phi }^{(\ell ')}$ ;

• • We have $\phi \le \phi '$ for all $\phi \in \Phi ^{\sqcup n}$ and $\phi ' \in \overline {\Phi }^{\sqcup n}$ .

To each $\alpha = \sum _{\phi \in \Phi } a_{\phi } \phi \in X^*(T)_n$ , we can assign several subsets $ P_{\alpha }$ of $\Phi ^{\sqcup n} \cup \overline {\Phi }^{\sqcup n}$

(3.4) $$ \begin{align} P_{\alpha}= \bigcup_{a_{\phi}> 0, ~ j = 1, \ldots, a_{\phi}} \{\phi^{(\ell_j)}\} \bigcup_{a_{\phi} < 0, ~ j = 1, \ldots, |a_{\phi}|} \{\overline{\phi^{(\ell_j)}}\} \end{align} $$

with all the $\ell _j$ ’s taking values in $\{1,\ldots ,n\}$ . If $n=1$ , then there is a unique way to assign a subset of $\Phi \cup \overline {\Phi }$ . When $n \ge 2$ , the assignment is not unique, and one possible choice is $P_{\alpha }:=P_{\alpha }^{(1)} \cup P_{\alpha }^{(2)} \cup \dots \cup P_{\alpha }^{(n)}$ with, for each $\ell \in \{1,\ldots ,n\}$ ,

$$ \begin{align*}P_{\alpha}^{(\ell)} =\{\phi \in \Phi^{(\ell)}: a_{\phi} \ge \ell\} \cup \{\overline{\phi} \in \overline{\Phi^{(\ell)}}: a_{\phi} \le -\ell \} \in {\mathcal P}(\Phi^{(\ell)} \cup \overline{\Phi^{(\ell)}}). \end{align*} $$

Proposition 3.3. Let $\alpha \in N \cap X^*(T)_n$ , and define $P_{\alpha } \subseteq \Phi ^{\sqcup n} \cup \overline {\Phi }^{\sqcup n}$ as above. Then the element

$$\begin{align*}[P_{\alpha}] := \bigwedge\nolimits_{\phi \in P_{\alpha}}\phi \end{align*}$$

is in $B(A^n)\otimes {\mathbb C}$ . Moreover, $B(A^n) \otimes {\mathbb C}$ (viewed as a ${\mathbb C}$ -vector space) has a basis consisting of $[P_{\alpha }]$ for all $\alpha \in N\cap X^*(T)_n$ and of $\phi ^{(\ell )} \wedge \overline {\phi ^{(\ell ')}}$ for all $\phi \in \Phi $ and $\ell ,\ell ' \in \{1,\ldots ,n\}$ .

Proof. For each $\alpha =\sum _{\phi \in \Phi } a_{\phi } \phi \in X^*(T)_n$ and each $P_{\alpha }$ of the form (3.4), we start by proving that

(3.5) $$ \begin{align} \alpha\in N \Longleftrightarrow \vert \sigma P_{\alpha} \cap \Phi^{\sqcup n} \vert=\vert \sigma P_{\alpha} \cap \overline{\Phi}^{\sqcup n} \vert \text{ for all }\sigma \in G\text{ and } \sum\nolimits_{\phi \in \Phi} a_{\phi} = 0. \end{align} $$

Consider the inner product $\langle \cdot , \cdot \rangle $ on $\bigoplus _{\phi \in \Phi \cup \overline {\Phi } } {\mathbb Z} \phi $ defined at the beginning of the proof of Lemma 3.2. Then for any $\sigma \in G$ , we have

$$ \begin{align*}\langle \sum_{\phi \in \Phi} a_{\phi} \phi , \sigma\sum_{\phi\in \Phi} (\phi - \overline\phi) \rangle = |P_{\alpha} \cap \sigma \Phi^{\sqcup n} | - |P_{\alpha} \cap \sigma \overline{\Phi}^{\sqcup n}|. \end{align*} $$

Recall that $X_*(\mathrm {MT}(A))$ is generated by $\mathrm {Gal}(E^c/{\mathbb Q})\sum _{\phi \in \Phi }\phi $ by Lemma 2.1. Hence

$$ \begin{align*} \sum_{\phi \in \Phi} a_{\phi} \phi \in N & \Longleftrightarrow \langle \sum_{\phi \in \Phi} a_{\phi} \phi , \sigma\sum_{\phi\in \Phi}\phi\rangle = 0 \text{ for all }\sigma \in G \\ & \Longleftrightarrow \langle \sum_{\phi \in \Phi} a_{\phi} \phi , \sigma\sum_{\phi\in \Phi} (\phi - \overline\phi) \rangle = 0 \text{ and } \langle \sum_{\phi \in \Phi} a_{\phi} \phi , \sigma\sum_{\phi\in \Phi} (\phi + \overline\phi) \rangle = 0 \text{ for all }\sigma \in G\\ & \Longleftrightarrow \langle \sum_{\phi \in \Phi} a_{\phi} \phi , \sigma\sum_{\phi\in \Phi} (\phi - \overline\phi) \rangle = 0 \text{ for all }\sigma \in G\text{ and } \langle \sum_{\phi \in \Phi} a_{\phi} \phi , \sum_{\phi \in \Phi\sqcup \overline{\Phi}}\phi \rangle = 0 \\ & \Longleftrightarrow \langle \sum_{\phi \in \Phi} a_{\phi} \phi , \sigma\sum_{\phi\in \Phi} (\phi - \overline\phi) \rangle = 0 \text{ for all }\sigma \in G\text{ and } \sum_{\phi \in \Phi} a_{\phi} = 0 \\ & \Longleftrightarrow |P_{\alpha} \cap \sigma \Phi^{\sqcup n} | - |P_{\alpha} \cap \sigma \overline{\Phi}^{\sqcup n}| = 0 \text{ for all }\sigma \in G\text{ and } \sum\nolimits_{\phi \in \Phi} a_{\phi} = 0. \end{align*} $$

Thus we have established (3.5). Now the last condition implies $[P_{\alpha }] \in B(A^n)\otimes {\mathbb C}$ by Pohlmann Theorem 3.1. So we have established the first assertion of the proposition.

Next by Theorem 3.1, the ${\mathbb C}$ -vector space $B(A^n) \otimes {\mathbb C}$ has a basis consisting of those $[P]$ , with $P \subseteq S^{\sqcup n} = \Phi ^{\sqcup n} \cup \overline {\Phi }^{\sqcup n}$ , such that $\vert \sigma P \cap \Phi ^{\sqcup n} \vert =\vert \sigma P \cap \overline {\Phi }^{\sqcup n} \vert $ for all $\sigma \in G$ . For each such P, define

$$\begin{align*}\alpha(P) := \sum_{\phi \in \Phi} \#\{\ell : \phi \in P\cap \Phi^{(\ell)}\} \phi - \sum_{\phi \in \Phi} \#\{\ell : \phi \in P\cap \overline{\Phi^{(\ell)}}\} \phi \in X^*(T)_n \end{align*}$$

Then by construction, there exists a subset $P_{\alpha (P)}$ of the form (3.4), assigned to $\alpha (P)$ , such that $P \setminus P_{\alpha (P)} = \{\phi _{i_1}^{(\ell _1)}, \overline {\phi _{i_1}^{(\ell ^{\prime }_1)}}, \ldots , \phi _{i_r}^{(\ell _r)},\overline {\phi _{i_r}^{(\ell ^{\prime }_r)}}\}$ for some $\phi _{i_1},\ldots ,\phi _{i_r} \in \Phi $ . Thus

$$\begin{align*}\vert \sigma P_{\alpha(P)} \cap \Phi^{\sqcup n} \vert = \vert \sigma P \cap \Phi^{\sqcup n} \vert - r = \vert \sigma P \cap \overline{\Phi}^{\sqcup n} \vert - r = \vert \sigma P_{\alpha(P)} \cap \overline{\Phi}^{\sqcup n} \vert \end{align*}$$

for all $\sigma \in G$ . On the other hand,

$$\begin{align*}\sum_{\phi\in \Phi}\#\{\ell : \phi \in P\cap \Phi^{(\ell)}\} = \vert P \cap \Phi^{\sqcup n} \vert=\vert P \cap \overline{\Phi}^{\sqcup n} \vert = \sum_{\phi\in\Phi}\#\{\ell : \phi \in P\cap \overline{\Phi^{(\ell)}}\}. \end{align*}$$

So $\alpha (P) \in N$ by (3.5). Now we can conclude.

4.1 CM Abelian Varieties of Weyl types

Let E be a CM field of degree $2g$ , and let $E^c$ be the Galois closure of E and G be the Galois group $G=\mathrm {Gal}(E^c/{\mathbb Q})$ . Let $\Phi $ be a CM type on E. Then G can be seen as a subgroup of $({\mathbb Z}/2{\mathbb Z})^g\rtimes S_g$ by [Reference Dodson.Dod84, Imprimitivity Theorem].

Assume E is of Weyl type. Then any CM type $\Phi '$ of E is conjugate of $\Phi $ by an element of G. Thus G acts transitively on the set of all CM types on E. This justifies that a CM field E being of Weyl type is independent of the choice of $\Phi $ . Moreover, a CM abelian variety of Weyl type is by definition simple, and thus the associated CM pair $(E,\Phi )$ is a primitive CM pair.

In the rest of this subsection, we will identify $S_g$ with $\{0\}\times S_g < G$ to ease notation.

Let F be the totally real field of degree g contained in E and let $F^c$ be the Galois closure of F. Then

(4.1) $$ \begin{align} \mathrm{Gal}(F^c/{\mathbb Q})=S_g \text{ and } \mathrm{Gal}(E^c/E)=\{0\}\times ({\mathbb Z}/2{\mathbb Z})^{g-1}\rtimes S_{g-1}. \end{align} $$

The Galois group G acts transitively on $\text {Hom} (E,{\mathbb C})=\Phi \cup \overline {\Phi }$ where $\Phi =\{\phi _1,\dots , \phi _g\}$ and $\overline {\Phi }= \{\overline {\phi }_1, \dots , \overline {\phi }_g \}$ . By convention $\phi _1$ is the identity map giving the inclusion $E\subseteq E^c\subseteq {\mathbb C}$ .

The action of $S_g$ preserves $\Phi $ and $\overline {\Phi }$ , and $S_g$ acts by permutation of the indexes of elements of $\Phi $ and of $\overline {\Phi }$ . Let $\{e_1,\ldots ,e_g\}$ be the canonical basis of $({\mathbb Z}/2{\mathbb Z})^g$ . Then the action of $({\mathbb Z}/2{\mathbb Z})^g$ on $\Phi $ is determined by

(4.2) $$ \begin{align} e_j \phi_k = \begin{cases} \overline{\phi}_j & \text{ if } k=j \\ \phi_k & \text{ if }k\not=j.\end{cases} \end{align} $$

In particular $\rho :=e_{\{1,\dots ,g\}}$ acts as the complex conjugation.

Let

(4.3) $$ \begin{align} H:= \text{Fix}(\phi_1)=\mathrm{Gal} (E^c/E)= \{0\}\times ({\mathbb Z}/2{\mathbb Z})^{g-1}\rtimes S_{g-1}. \end{align} $$

Let $\sigma =(1,2,3,\dots ,g)$ be the standard g-cycle in $S_g$ . Then

$$\begin{align*}G/H = \{H, \sigma H, \sigma^2 H, \dots, \sigma^{g-1} H, \rho H, \sigma\rho H, \dots, \sigma^{g-1} \rho H\} \end{align*}$$

and G acts on $G/H$ by multiplication on the left. We have a natural bijection

(4.4) $$ \begin{align} \eta \colon \text{Hom}(E,{\mathbb C}) \xrightarrow{\sim} G/H \end{align} $$

which sends $\phi _j\mapsto \sigma ^{j-1}H$ and $\overline {\phi }_j \mapsto \sigma ^{j-1}\rho H$ (for all $j \in \{1,\ldots ,g\}$ ).

Proof. It suffices to check that $\eta $ is equivariant for all $\sum _{k\in I}e_k$ with $I \subseteq \{1,\ldots ,g\}$ and for all $\alpha \in S_g$ .

We start with $\sum _{k\in I}e_k$ . We have seen that $(\sum _{k\in I}e_k) \cdot \phi _j$ equals $\overline {\phi }_j$ if $j \in I$ and equals $\phi _j$ otherwise. On the other hand

$$\begin{align*}(\sum_{k\in I}e_k) \sigma^{j-1} H= \sigma^{j-1} (\sum_{k\in \sigma^{1+g-j}(I)}e_k) H= \begin{cases} \sigma^{j-1} H & \text{ if } 1\notin \sigma^{1+g-j}(I) \\ \sigma^{j-1} \rho H & \text{ if } 1\in \sigma^{1+g-j}(I). \end{cases} \end{align*}$$

The condition $1\not \in \sigma ^{1+g-j}(I)$ is equivalent to $\sigma ^{j-1}(1)=j\notin I$ . Hence we are done for elements in $({\mathbb Z}/2{\mathbb Z})^g$ .

Next take $\alpha \in S_g$ . Then $\alpha \cdot \phi _j = \phi _{\alpha (j)}$ . On the other hand, denote by $\beta = \alpha \sigma ^{j-1}$ . Then $\beta \in \sigma ^{j_\beta -1} H$ for some $j_{\beta }$ . Thus $\alpha = \sigma ^{j_{\beta }-1} h \sigma ^{1+g-j}$ for some $h \in H$ , and hence $ \alpha (j) = \sigma ^{j_{\beta }-1} h(1) = \sigma ^{j_{\beta }-1}(1) = j_{\beta }$ . So

$$\begin{align*}\alpha \sigma^{j-1}H = \beta H = \sigma^{j(\alpha)-1} H. \end{align*}$$

We are done.

4.2 CM abelian varieties of anti-Weyl type

Let $(E,\Phi )$ be a CM pair with E of Weyl type, and let $(E^*,\Phi ^*)$ be its reflex CM pair. The stabilizer of $\Phi $ is the subgroup $H'=\{0\}\times S_g$ of $G = \mathrm {Gal}(E^c/{\mathbb Q})$ , therefore $E^* = (E^c)^{H'}$ . Now we compute the reflex CM type $\Phi ^*$ on $E^*$ .

For each $I \subseteq \{1,\ldots ,g\}$ , denote by $\varepsilon _I := \sum _{j \in I} e_j$ . Then the action of G on $\text {Hom}(E^*,{\mathbb C})$ gives a G-equivariant bijection

(4.5) $$ \begin{align} \eta^* \colon \text{Hom} (E^*,{\mathbb C}) \xrightarrow{\sim} G/ H' =\{f H'\}_{f\in ({\mathbb Z}/2{\mathbb Z})^g}= \{ \varepsilon_I H' \}_{I\subseteq\{1,\dots,g\}}. \end{align} $$

By abuse of notation, let us identify $\text {Hom}(E^*,{\mathbb C})$ with $G/ H' =\{f H'\}_{f\in ({\mathbb Z}/2{\mathbb Z})^g}= \{ \varepsilon _I H' \}_{I\subseteq \{1,\dots ,g\}}$ via the bijection $\eta ^*$ . As $X^*((E^*)^{\times })= {\mathbb Z}[\text {Hom}(E^*,{\mathbb C})]$ , we denote by $\varepsilon _I$ the basis of $X^*((E^*)^{\times })$ corresponding to the coset $ \varepsilon _I H'$ .

Let A (resp. $A^*$ ) be a CM abelian variety associated with the CM pair $(E,\Phi )$ (resp. with $(E^*,\Phi ^*)$ ). Then A is of Weyl type and $A^*$ is of anti-Weyl type. Let T be the maximal torus of $\mathrm {GSp}(H^1(A,{\mathbb Q}))$ described in §2.3. Then it is easy to check that $\mathrm {MT}(A)$ is T. Recall that $(E,\Phi )$ is primitive, so by the last paragraph of §2.4, the reflex of $(E^*, \Phi ^*)$ is $(E,\Phi )$ and $\mathrm {MT}(A^*) \simeq \mathrm {MT}(A)$ . So $\dim \mathrm {MT}(A^*)$ attains its lower bound $\log _2 \dim A^* +2$ . This makes $A^*$ interesting from the point of view of Hodge theory.

We end this section with the following proposition on the reflex norm. Recall that we have identified $\text {Hom}(E^*,{\mathbb C})$ with $G/H'$ via $\eta ^*$ above. So $X^*((E^*)^{\times })={\mathbb Z}[\text {Hom}(E^*,{\mathbb C})]$ is identified with ${\mathbb Z}[G/H']$ .

Proposition 4.5. The group homomorphism induced by the reflex norm (2.4) (but with E and $E^*$ switched)

$$\begin{align*}\mathrm{rec}^*\colon X^*((E^*)^{\times})={\mathbb Z}[G/H']\longrightarrow X^*(E^\times)={\mathbb Z}[\text{Hom}(E,{\mathbb C})] \end{align*}$$

is given by: for all $I\subseteq \{1,\dots ,g\}$ ,

$$\begin{align*}\varepsilon_I H' \mapsto \sum_{j \not\in I} \phi_j + \sum_{j \in I} \overline{\phi}_j. \end{align*}$$

Proof. The reflex norm (2.4) applied to our situation is

$$\begin{align*}\mathrm{rec} \colon E^\times \rightarrow (E^*)^\times, \qquad a \mapsto \prod_{\phi \in \Phi} \phi(a). \end{align*}$$

Since $\Phi = \{\phi _1,\ldots ,\phi _g\}$ , we have

$$\begin{align*}\mathrm{rec}^*( \varepsilon_I H' )= \sum_{j=0}^{g-1} \varepsilon_I \phi_j. \end{align*}$$

Hence the result follows from (4.2).

6.1 Basic definitions

Let $\{e_1,\ldots ,e_g\}$ be the canonical basis of $({\mathbb Z}/2{\mathbb Z})^g$ . We have bijections

(6.1) $$ \begin{align} \begin{array}{ccccc} {\mathcal P}(\{1,\ldots,g\}) & \simeq & ({\mathbb Z}/2{\mathbb Z})^g & \simeq & \{\text{CM types of } E\}\\ I & \mapsto & \varepsilon_I:=\sum_{j\in I}e_j & \mapsto & \{\phi_j : j\not\in I\} \cup \{\overline{\phi}_j : j \in I\}. \end{array} \end{align} $$

The first bijection is natural and canonical, and the second depends on the choice of the CM type $\Phi _E$ . The group G acts on each of the three sets via its natural action on $({\mathbb Z}/2{\mathbb Z})^g$ . Here is the formula for the induced action on ${\mathcal P}(\{1,\ldots ,g\})$ : For each $I \subseteq \{1,\ldots ,g\}$ and each $\theta \in G$ , we have

(6.2) $$ \begin{align} \theta\cdot I = \beta_{\theta}^{-1} (I \bigvee I_{\theta}) \qquad \text{for any }\theta = (\varepsilon_{I_{\theta}}, \beta_{\theta}) \in G < ({\mathbb Z}/2{\mathbb Z})^g \rtimes S_g \end{align} $$

where $I \bigvee I_{\theta } = (I \cup I_{\theta }) \setminus (I \cap I_{\theta }) \subseteq \{1,\ldots ,g\}$ and the right hand side is the usual action of $S_g$ on $\{1,\ldots ,g\}$ . The induced action of G on $\{\text {CM types of }E\}$ is the one from [Reference Dodson.Dod84, $\S$ 1.2, Proposition].

Let $O_1,\ldots ,O_m$ be the G-orbits, and up to renumbering assume $O_1$ is the orbit of $\emptyset $ (the orbit of $(0,\ldots ,0)$ , or the orbit of $\Phi _E$ ).

We construct a CM abelian variety from the sets in (6.1) as follows. Let

• • $S := {\mathcal P}(\{1,\ldots ,g\})$ with the action of G as above;

• • $\boldsymbol {\Phi } := \{ I \subseteq \{1,\ldots , g\} : 1\not \in I\}$ and $\overline {\boldsymbol {\Phi }} := \{I \subseteq \{1,\ldots ,g\} : 1\in I\}$ .

Then $\text {Hom}_G(S, E^c)$ is a CM algebra split by $E^c$ , i.e. it is isomorphic to the product of some CM fields in $E^c$ . Indeed, we have an isomorphism

$$ \begin{align*}\text{Hom}_G(S, E^c)\simeq (E^c)^{\mathrm{Stab}_G(\Phi_1)} \times \cdots \times (E^c)^{\mathrm{Stab}_G(\Phi_m)}, \end{align*} $$

with some $\Phi _1 \in O_1, \ldots , \Phi _m \in O_m$ , and each $(E^c)^{\mathrm {Stab}_G(\Phi _r)}$ is the reflex field of $(E,\Phi _r)$ and hence is CM.

Note that $S = \text {Hom}( \text {Hom}_G(S,E^c), {\mathbb C})$ . Hence $\boldsymbol {\Phi }$ defines a CM type on $\text {Hom}_G(S, E^c)$ . If we look at S as the set of CM types on E under the identification (6.1), then $\boldsymbol {\Phi }$ consists of those CM types which contains $\phi _1$ .

Let ${\mathcal O}$ be the ring of integers of $\text {Hom}_G(S, E^c)$ .

Definition 6.1. A CM abelian variety is said to be of generalized anti-Weyl type arising from $(E,\Phi _E)$ , if it is isogeneous to

$$\begin{align*}{\mathbb C}^{|S|/2} / \boldsymbol{\Phi}({\mathcal O}) \end{align*}$$

for S, $\boldsymbol {\Phi }$ , ${\mathcal O}$ above.

By definition, up to isogeny there exists a unique CM abelian variety of generalized anti-Weyl type arising from each CM pair $(E,\Phi _E)$ , which we denote by $\mathcal {A}(E,\Phi _E)$ . It is clear that $\dim \mathcal {A}(E,\Phi _E) = 2^{[E:{\mathbb Q}]/2 -1}$ .

For each $r \in \{1,\ldots ,m\}$ , set $\boldsymbol {\Phi }_r := \boldsymbol {\Phi }\cap O_r$ . As discussed above, each $\text {Hom}_G(O_r, E^c)$ is a CM field, and we have a natural identification $O_r = \text {Hom}(\text {Hom}_G(O_r, E^c), {\mathbb C})$ . Now $\boldsymbol {\Phi }_r$ is a CM type on $\text {Hom}_G(O_r, E^c)$ . Let ${\mathcal O}_r$ be the ring of integers of $\text {Hom}_G(O_r, E^c)$ . Then by construction $\mathcal {A}(E,\Phi _E)$ is isogeneous to

(6.3) $$ \begin{align} {\mathbb C}^{|O_1|/2}/\boldsymbol{\Phi}_1({\mathcal O}_1) \times\cdots \times {\mathbb C}^{|O_m|/2}/\boldsymbol{\Phi}_m({\mathcal O}_m). \end{align} $$

Each ${\mathbb C}^{|O_r|/2}/\boldsymbol {\Phi }_r({\mathcal O}_r)$ is a CM abelian variety associated with the CM pair $(\text {Hom}_G(O_r, E^c), \boldsymbol {\Phi }_r)$ .

As above, up to renumbering we always assume $O_1$ to be the orbit of $\emptyset $ (the orbit of $(0,\ldots ,0)$ or the orbit of $\Phi _E$ under the identification (6.1)).

A main result of this section is to prove that the first compagnon is precisely A up to isogeny (Proposition 6.5). As a consequence, any simple CM abelian variety is, up to isogeny, an abelian subvariety of a CM abelian variety of generalized anti-Weyl type.

6.2 The Galois action on $\text {Hom}(E,{\mathbb C})$

Use the notation from §6.1. We fix some further notation for the computation. Let

$$ \begin{align*}H:= \mathrm{Fix}(\phi_1) = \mathrm{Gal}(E^c/E) = G \cap (\{0\} \times ({\mathbb Z}/2{\mathbb Z})^{g-1}\rtimes S_{g-1}). \end{align*} $$

Then $|G/H| = [E:{\mathbb Q}] = 2g$ .

For each $j \in \{1,\ldots ,g\}$ , there exists $\alpha _j = (\varepsilon _{I_j}, \beta _j) \in G < ({\mathbb Z}/2{\mathbb Z})^g \rtimes S_g$ such that $\beta _j(1) = j$ . This is because the image of $G<({\mathbb Z}/2{\mathbb Z})^g \rtimes S_g \rightarrow S_g$ acts transitively on $\{1,\ldots ,g\}$ . Up to replacing $\alpha _j$ by $\rho \circ \alpha _j$ , we may and do assume that $j\not \in I_j$ for each $j\in \{1,\ldots ,g\}$ . Now

$$\begin{align*}G/H = \{\alpha_1 H, \ldots, \alpha_g H, \alpha_1 \rho H, \ldots, \alpha_g \rho H\} \end{align*}$$

since $|G/H| = 2g$ . We can define a bijection

(6.4) $$ \begin{align} \eta \colon \text{Hom}(E,{\mathbb C}) \xrightarrow{\sim} G/H \end{align} $$

by sending $\phi _j \mapsto \alpha _j H$ and $\overline {\phi }_j\mapsto \alpha _j \rho H$ for each $j \in \{1,\ldots ,g\}$ .

As for the case of Weyl type CM abelian varieties, we prove the following lemma. Notice that this gives a new proof of Lemma 4.2.

6.3 The first Galois orbit $O_1$

Recall that $O_1$ is the G-orbit of $\emptyset $ in ${\mathcal P}(\{1,\ldots ,g\})$ ; under the identifications (6.1) it is the orbit of $(0,\ldots ,0)$ in $({\mathbb Z}/2{\mathbb Z})^g$ , or the orbit of $\Phi _E$ in $\{\text {CM types of }E\}$ .

Proof. Denote for simplicity $H' = \mathrm {Stab}_G(\Phi _E)$ . Since $\Phi _E$ corresponds to

$$ \begin{align*}\mathbf{0} = (0,\ldots,0) \in ({\mathbb Z}/2{\mathbb Z})^g \end{align*} $$

under (6.1), we have $H' = \mathrm {Stab}_G(\mathbf {0}) = G \cap (\{0\}\times S_g)$ .

We have $\text {Hom}_G(O_1,E^c) \simeq (E^c)^{H'}$ and as

$$ \begin{align*}\text{Hom}(\text{Hom}_G(O_1,E^c),{\mathbb C}) = O_1 \simeq G/H', \end{align*} $$

the reflex CM type $\Phi _E^*$ equals $\{ \theta ^{-1} H' : \theta \in G \text { with } \theta H \in \Phi _E\}$ . We want to prove that

$$ \begin{align*}\Phi_E^* = \boldsymbol{\Phi}_1 \subseteq O_1. \end{align*} $$

By Lemma 4.2, the condition $\theta H \in \Phi _E$ is equivalent to $\theta H = \alpha _j H$ for some j, and hence equivalent to $\theta = (\varepsilon _{I_j}, \beta _j)(\varepsilon _I , h')$ for some I not containing $1$ , some $h' \in S_{g-1}$ such that $(\varepsilon _I, h') \in G$ , and some j. Then $\theta ^{-1} = (\varepsilon _{h^{\prime -1}(I)}, h^{\prime -1}) (\varepsilon _{\beta ^{-1}_j(I_j)}, \beta _j^{-1}) = (\varepsilon _{h^{\prime -1}(I)} + \varepsilon _{h^{\prime -1}\beta ^{-1}_j(I_j)} , h^{\prime -1}\beta _j^{-1})$ . Let $I_{\theta ^{-1}} \subseteq \{1,\ldots , g\}$ be such that $\varepsilon _{I_{\theta ^{-1}}} = \varepsilon _{h^{\prime -1}(I)} + \varepsilon _{h^{\prime -1}\beta ^{-1}_j(I_j)}$ . Then $1 \not \in I_{\theta ^{-1}}$ because $1 \not \in I$ , $1 \not \in \beta ^{-1}_j(I_j)$ and $h' \in S_{g-1}$ .

In summary, the reflex norm $\Phi _E^*$ is contained in $\boldsymbol {\Phi }_1 = \boldsymbol {\Phi } \cap O_1 \subseteq O_1 \simeq G/H'$ . But $|\Phi _E^*| = |\boldsymbol {\Phi }_1| = |O_1|/2$ . Hence $\Phi _E^* = \boldsymbol {\Phi }_1$ . We are done.

Here are two immediate corollaries.

6.4 Powers of CM abelian varieties of generalized anti-Weyl type

Let $A:={\mathcal A}(E,\Phi _E)$ be an abelian variety of generalized anti-Weyl type arising from $(E,\Phi _E)$ . Later on we will study an arbitrary power of ${\mathcal A}(E,\Phi _E)$ . In this subsection, we make a preliminary discussion and we fix some notations.

Recall that A is associated with the CM pair $\left ( \text {Hom}_G(S,E^c), \boldsymbol {\Phi } \right )$ , with $S = {\mathcal P}(\{1,\ldots ,g\})$ and $\boldsymbol {\Phi } = \{I \subseteq \{1,\ldots ,g\} : 1\not \in I\}$ . Using the notation of $\S $ 2.2, for each $n \ge 1$ , we can construct a CM pair as follows. The CM algebra is $ \text {Hom}_G(S,E^c)^n$ , with $\text {Hom} ( \text {Hom}_G(S,E^c)^n, {\mathbb C}) = S^{\sqcup n}$ ; the CM type is $\boldsymbol {\Phi }^{\sqcup n}$ .

The CM algebra $\text {Hom}_G(S,E^c)$ acts on the space of holomorphic $1$ -forms (resp. anti-holomorphic $1$ -forms) on A via $\boldsymbol {\Phi }$ (resp. via $\overline {\boldsymbol {\Phi }}$ ), and hence we have an eigenbasis

$$\begin{align*}\{\varepsilon_I\}_{I\subseteq \{1,\dots,g\}, 1\notin I} , \quad (\text{resp. }\{\varepsilon_I\}_{I\subseteq \{1,\dots,g\}, 1\in I}) \end{align*}$$

for this action, with $\varepsilon _I$ an eigenvector associated with $I \in \boldsymbol {\Phi }$ (resp. with $I \in \overline {\boldsymbol {\Phi }}$ ).

For each $\ell \in \{1,\ldots ,n\}$ , let

$$\begin{align*}I^{(\ell)}, \quad \text{(resp. }\varepsilon_I^{(\ell)} \text{)} \end{align*}$$

denote the subset $I \in {\mathcal P}(\{1,\ldots ,g\})$ on the $\ell $ -th copy of ${\mathcal P}(\{1,\ldots ,g\})^{\sqcup n}$ (resp. the $1$ -form $\varepsilon _I$ on the $\ell $ -th factor of $A^n$ ).

We fix a partial order on $S = {\mathcal P}(\{1,\ldots ,g\})$ such that $I \le J$ if $1\notin I$ and $1\in J$ and $I \le J \Rightarrow J^{\mathrm {c}} \le I^{\mathrm {c}}$ . Then it induces a partial order on $S^{\sqcup n}$ such that $1\le \ell _1\le \ell _2 \le n \Rightarrow I^{(\ell _1)} \le J^{(\ell _2)}$ .

7.1 Proof of our main theorem

We start by explaining how to deduce Theorem 1.1 from Theorem 7.2. The notations are different from the rest of this section.

Let A be a simple CM abelian variety, associated with the CM pair $(E,\Phi )$ . Its reflex $(E^*, \Phi ^*)$ is a CM pair with $E^*$ a field. Let ${\mathcal A}(E^*,\Phi ^*)$ be the generalized anti-Weyl CM abelian variety arising from $(E^*,\Phi ^*)$ . By Corollary 6.6, up to isogeny ${\mathcal A}(E^*,\Phi ^*) = A \times B$ for some CM abelian variety B. By construction of ${\mathcal A}(E^*,\Phi ^*)$ , we see that B is split over $E^c$ . Theorem 1.1 then follows from Theorem 7.2 and the comment below (that Hodge relations in degree $1$ do not give algebraic relations between the holomorphic periods).

7.2 A first discussion on Hodge cycles on generalized anti-Weyl CM abelian varieties

Proof. Recall $S = {\mathcal P}(\{1,\ldots ,g\})$ . We wish to apply Theorem 3.1 (with S being our $S^{\sqcup n}$ and $\Phi $ being our $\boldsymbol {\Phi }^{\sqcup n}$ ). Thus the set P from Theorem 3.1 is in our case of the form $\{I_1^{(\ell _1)},\ldots ,I_{2p}^{(\ell _{2p})}\}$ with $I_j^{(\ell _j)} \in {\mathcal P}(\{1,\ldots ,g\})$ .

Recall that $\boldsymbol {\Phi } = \{I \in {\mathcal P}(\{1,\ldots ,g\}) : 1 \not \in I\}$ and $\overline {\boldsymbol {\Phi } } = \{I \in {\mathcal P}(\{1,\ldots ,g\}) : 1 \in I\}$ . Hence $\vert P\cap \boldsymbol {\Phi }^{\sqcup n} \vert =\vert P\cap \overline {\boldsymbol {\Phi } }^{\sqcup n}\vert $ is equivalent to $1$ being contained in exactly half of the sets $I_1^{(\ell _1)},\ldots ,I_{2p}^{(\ell _{2p})}$ .

The condition given by the equation (3.2) is in our situation $\vert P\cap \sigma \boldsymbol {\Phi }^{\sqcup n} \vert =\vert P\cap \sigma \overline {\boldsymbol {\Phi } }^{\sqcup n}\vert $ for all $\sigma \in G < ({\mathbb Z}/2{\mathbb Z})^2 \rtimes S_g$ . For the canonical basis $\{e_1,\ldots ,e_g\}$ of $({\mathbb Z}/2{\mathbb Z})^g$ , we have $e_1\cdot \boldsymbol {\Phi }^{\sqcup n} = \overline {\boldsymbol {\Phi }}^{\sqcup n}$ and $e_1 \cdot \overline {\boldsymbol {\Phi }}^{\sqcup n} = \boldsymbol {\Phi }^{\sqcup n}$ , and $e_j \cdot \boldsymbol {\Phi }^{\sqcup n} = \boldsymbol {\Phi }^{\sqcup n} $ and $e_j \cdot \overline {\boldsymbol {\Phi } }^{\sqcup n} = \overline {\boldsymbol {\Phi } }^{\sqcup n}$ for all $j \ge 2$ . Hence (3.2) is further equivalent to $\vert P\cap \sigma \boldsymbol {\Phi }^{\sqcup n} \vert =\vert P\cap \sigma \overline {\boldsymbol {\Phi } }^{\sqcup n}\vert $ for all $\sigma \in \mathrm {im}(G \rightarrow S_g)$ . To ease notation, for each $j \in \{1,\ldots , g\}$ , set $\boldsymbol {\Phi }_j := \{I \subseteq \{1,\ldots , g\} : j \not \in I\}$ and $\overline {\boldsymbol {\Phi }_j} := \{I \subseteq \{1,\ldots , g\} : j \in I\}$ . We have

$$\begin{align*}\{\sigma\boldsymbol{\Phi}^{\sqcup n} : \sigma \in G \} \simeq \{\boldsymbol{\Phi}_j^{\sqcup n} : j =1, \ldots,g \} \quad \text{and} \quad \{\sigma\overline{\boldsymbol{\Phi} }^{\sqcup n} : \sigma \in G\} \simeq \{\overline{\boldsymbol{\Phi}_j}^{\sqcup n} : j = 1,\ldots, g\} \end{align*}$$

as $\mathrm {im}(G \rightarrow S_g)$ acts transitively on $\{1,\ldots ,g\}$ . Therefore the condition (3.2) is equivalent to: $\vert P\cap \boldsymbol {\Phi }_j^{\sqcup n}\vert =\vert P\cap \overline {\boldsymbol {\Phi }_j}^{\sqcup n}\vert $ for all $j \in \{1,\ldots ,g\}$ . As for the case $j=1$ discussed above, this is equivalent to: each $j \in \{1,\ldots ,g\}$ is contained in exactly half of the sets $I_1^{(\ell _1)},\ldots ,I_{2p}^{(\ell _{2p})}$ . Hence we are done.

Proposition 7.3 yields immediately the following corollary.

7.3 $(2,2)$ -Hodge cycles on generalized anti-Weyl CM abelian varieties

As indicated by Theorem 7.2, $(2,2)$ -Hodge cycles play a particularly important role in the Hodge rings of generalized anti-Weyl CM abelian varieties. It is therefore important to understand the Galois action on these $(2,2)$ -Hodge cycles. We will do this in the current subsection. We will then prove Theorem 7.1 from the description of this Galois action. Notice that one can also prove Theorem 7.1 directly via Proposition 7.3 without analysing the Galois action on the $(2,2)$ -cycles.

Recall the formula for the action of G on ${\mathcal P}(\{1,\ldots ,g\})$ from (6.2):

$$\begin{align*}\theta \cdot M = \beta_{\theta}^{-1} (M \bigvee I_{\theta}) \qquad\!\! \text{for any }M\subseteq \{1,\ldots,g\}\text{ and any }\theta = (\varepsilon_{I_{\theta}}, \beta_{\theta}) \in G < ({\mathbb Z}/2{\mathbb Z})^g \rtimes S_g, \end{align*}$$

with $M \bigvee I_{\theta } = (M \cup I_{\theta }) \setminus (M \cap I_{\theta }) \subseteq \{1,\ldots ,g\}$ and the right-hand side is the usual action of $S_g$ on $\{1,\ldots ,g\}$ . Notice that this action extends to an action of the whole $({\mathbb Z}/2{\mathbb Z})^g\rtimes S_g$ on ${\mathcal P}(\{1,\ldots ,g\})$ by the same formula.

Here is the Galois action on the $(2,2)$ -forms. For each $\theta \in G$ and any $I, J, K, L \subseteq \{2,\ldots ,g\}$ , we have

(7.3) $$ \begin{align} \theta(\varepsilon_I^{(\ell_1)} \wedge \varepsilon_J^{(\ell_2)} \wedge \varepsilon_{K^{\mathrm{c}}}^{(\ell_3)} \wedge \varepsilon_{L^{\mathrm{c}}}^{(\ell_4)}) = \varepsilon_{\theta\cdot I}^{(\ell_1)} \wedge \varepsilon_{\theta\cdot J}^{(\ell_2)} \wedge \varepsilon_{\theta\cdot K^{\mathrm{c}}}^{(\ell_3)} \wedge \varepsilon_{\theta\cdot L^{\mathrm{c}}}^{(\ell_4)}. \end{align} $$

We start with the following lemma. The computation involved in its proof is useful.

Lemma 7.5. Let $I, J, K,L$ be subsets of $\{1,\dots ,g\}$ such that

$$\begin{align*}I\cup J=K\cup L \text{ and } I\cap J=K\cap L. \end{align*}$$

Then

$$\begin{align*}(\theta\cdot I)\cup (\theta\cdot J)=(\theta\cdot K)\cup (\theta\cdot L) \text{and } (\theta\cdot I)\cap (\theta\cdot J)=(\theta\cdot K)\cap (\theta\cdot L) \end{align*}$$

for all $\theta \in ({\mathbb Z}/2{\mathbb Z})^g \rtimes S_g$ (and in particular for all $\theta \in G$ ).

Proof. As $\theta = \left (\prod _{j\in I_{\theta }}(\varepsilon _{\{j\}},1)\right )(0,\beta _{\theta })$ , it suffices to prove the result for $(0,\beta _{\theta })$ and for $e_j=\varepsilon _{\{j\}}$ for each $j\in \{1,\ldots ,g\}$ .

The expected conclusion easily holds true for $(0,\beta _{\theta })$ , which gives a bijection on $\{1,\ldots ,g\}$ . Hence it remains to check for $e_j=\varepsilon _{\{j\}}$ for each $j\in \{1,\dots ,g\}$ . There are 3 cases up to the symmetries.

• • (a) $j\in I$ , $j\notin J$ , $j\in K$ , $j\notin L$ .

• • (b) $j\in I\cap J=K\cap L$ .

• • (c) $j\in (I\cup J)^{\mathrm {c}}=(K\cup L)^{\mathrm {c}}$ .

In case (a), we have

(7.4) $$ \begin{align} e_j\cdot I = I\setminus\{j\}, ~ e_j\cdot J=J\cup \{j\}, ~ e_j\cdot K=K\setminus\{j\}, ~ e_j\cdot L=L\cup\{j\}. \end{align} $$

Hence $(e_j\cdot I)\cup (e_j\cdot J) = I\cup J = K\cup L = (e_j\cdot K)\cup (e_j\cdot L)$ and $(e_j\cdot I) \cap (e_j\cdot J) = I\cap J = K\cap L = (e_j\cdot K)\cap (e_j\cdot L)$ .

In case (b), we have

(7.5) $$ \begin{align} (e_j\cdot I)=I\setminus\{j\}, ~e_j\cdot J=J\setminus\{j\},~ e_j\cdot K=K\setminus\{j\}, ~e_j\cdot L=L\setminus\{j\}. \end{align} $$

Therefore $ (e_j\cdot I)\cup (e_j\cdot J)= (I\cup J)\setminus \{j\}=(K\cup L)\setminus \{j\}=(e_j\cdot K)\cup (e_j\cdot L) $ and $ (e_j\cdot I)\cap (e_j\cdot J)= (I\cap J)\setminus \{j\}=(K\cap L)\setminus \{j\}=(e_j\cdot K)\cap (e_j\cdot L) $ .

In case (c), we have

(7.6) $$ \begin{align} e_j\cdot I=I\cup \{j\}, ~e_j\cdot J=J\cup \{j\}, ~e_j\cdot K=K\cup \{j\}, ~e_j\cdot L=L\cup \{j\}. \end{align} $$

Therefore $ (e_j\cdot I)\cup (e_j\cdot J)= (I\cup J)\cup \{j\}=(K\cup L)\cup \{j\}=(e_j\cdot K)\cup (e_j\cdot L) $ and $ (e_j\cdot I)\cap (e_j\cdot J)= (I\cap J)\cup \{j\}=(K\cap L)\cup \{j\}=(e_j\cdot K)\cap (e_j\cdot L) $ .

The key point to prove Theorem 7.1 is the following lemma.

Now we are ready to finish the proof of Theorem 7.1.

Proof of Theorem 7.1.

By Pohlmann’s Theorem 3.1, $B^2(A^n)\otimes {\mathbb C}$ has a basis consisting of elements of the form $\varepsilon _I^{(\ell _1)}\wedge \varepsilon _J^{(\ell _2)}\wedge \varepsilon _{K^{\mathrm {c}}}^{(\ell _3)}\wedge \varepsilon _{L^{\mathrm {c}}}^{(\ell _4)}$ with $I,J,K,L \subseteq \{2,\ldots ,g\}$ such that for all $\theta \in G$ , we have the following property: the form

$$\begin{align*}\varepsilon_{\theta\cdot I}^{(\ell_1)} \wedge \varepsilon_{\theta\cdot J}^{(\ell_2)} \wedge \varepsilon_{\theta\cdot K^{\mathrm{c}}}^{(\ell_3)} \wedge \varepsilon_{\theta\cdot L^{\mathrm{c}}}^{(\ell_4)} \end{align*}$$

is in $B^2(A^n)\otimes {\mathbb C}$ , i.e. $1$ appears in exactly $2$ of the sets $(\theta \cdot I), (\theta \cdot J),(\theta \cdot K^{\mathrm {c}}),(\theta \cdot L^{\mathrm {c}})$ . By Lemma 7.6, this occurs if and only if $I\cup J=K\cup L$ and $I\cap J=K\cap L$ . Hence we are done.

7.4 Hodge relations between periods of generalized anti-Weyl CM abelian varieties

We start with the following lemma.

Lemma 7.7. The Hodge relations generated by $(2,2)$ -Hodge cycles on A contain the relation

$$\begin{align*}\Theta_I \Theta_{\emptyset}^{r-1} = \Theta_{\{i_1\}} \cdots \Theta_{\{i_r\}} \end{align*}$$

for each $I = \{i_1,\ldots ,i_r\} \subseteq \{1,\ldots ,g\}$ with $r \ge 2$ .

Proof. The proof is inspired by the proof of Corollary 5.2.

Let us do induction on r. When $r=2$ , by Theorem 7.1 we have that $\varepsilon _I \wedge \varepsilon _{\emptyset } \wedge \varepsilon _{\{i_1\}^{\mathrm {c}}} \wedge \varepsilon _{\{i_2\}^{\mathrm {c}}}$ is a $(2,2)$ -Hodge cycle (base changed to ${\mathbb C}$ ) on A. Hence we obtain the desired Hodge relation $\Theta _I \Theta _{\emptyset } = \Theta _{\{i_1\}}\Theta _{\{i_2\}}$ .

Assume the result is proved for $r-1 \ge 2$ . Take I with $|I|=r \ge 3$ . Then by applying Theorem 7.1 to A and the quadruple $I, \emptyset , \{i_1\}, I\setminus \{i_1\}$ , we obtain a Hodge relation in degree $2$

$$\begin{align*}\Theta_I \Theta_{\emptyset} = \Theta_{\{i_1\}}\Theta_{I\setminus\{i_1\}}. \end{align*}$$

By induction hypothesis applied to $I\setminus \{i_1\}$ , whose cardinality is $r-1 \ge 2$ , we get a Hodge relation generated by those given by $(2,2)$ -Hodge cycles on A

$$\begin{align*}\Theta_{I \setminus \{i_1\}} \Theta_{\emptyset}^{ r-2} = \Theta_{\{i_2\}} \cdots \Theta_{\{i_r\}}. \end{align*}$$

Thus we can conclude by multiplying both Hodge relations above.

Now we are ready to finish the proof of Theorem 7.2.

Proof of Theorem 7.2.

The conclusion trivially holds true if $g=1$ , so we assume $g\ge 2$ .

Consider a Hodge relation given by a $(p,p)$ -Hodge cycle on $A^n$ with $p \ge 3$ and $n \ge 1$ . By Proposition 7.3, this $(p,p)$ -Hodge cycle is a linear combination of cycles of the form

$$\begin{align*}\alpha = \varepsilon_{I_1}^{(\ell_1)} \wedge \cdots \wedge \varepsilon_{I_{2p}}^{(\ell_{2p})} \end{align*}$$

for subsets $I_1^{(\ell _1)} ,\ldots ,I_{2p}^{(\ell _{2p})} \subseteq \{1,\ldots ,g\}$ with each $j \in \{1,\ldots ,g\}$ appearing exactly p times in $I_1^{(\ell _1)},\ldots ,I_{2p}^{(\ell _{2p})}$ . In particular, $\sum _{l=1}^{2p} |I_l^{(\ell _l)}| = pg$ .

Without loss of generality, we assume that $1 \not \in I_1^{(\ell _1)},\ldots ,I_p^{(\ell _p)}$ and $1 \in I_{p+1}^{(\ell _{p+1})},\ldots ,I_{2p}^{(\ell _{2p})}$ . Thus the Hodge relation given by $\alpha $ is

(7.7) $$ \begin{align} \Theta_{I_1}\cdots \Theta_{I_p} = \Theta_{I_{p+1}^{\mathrm{c}}}\cdots \Theta_{I_{2p}^{\mathrm{c}}} \end{align} $$

with all periods being holomorphic.

We may assume that $I_l^{(\ell _l)} \not = (I_{l'}^{(\ell _{l'})})^{\mathrm {c}}$ for any $l, l' \in \{1,\ldots ,2p\}$ . This is equivalent to saying that $\alpha $ is not the wedge product of a $(1,1)$ -cycle with a $(p-1,p-1)$ -cycle $\alpha '$ .

Applying Lemma 7.7 to each $I_1^{(\ell _1)},\ldots ,I_{2p}^{(\ell _{2p})}$ (if $|I_l^{(\ell _l)}|=1$ then we do nothing), we obtain $2p$ equalities up to $\overline {\mathbb {Q}}^*$ which are either trivial equalities or Hodge relations generated by those in degree $2$ . Multiplying these $2p$ equalities, we then have by Proposition 7.3

$$\begin{align*}\Theta_{I_1} \cdots \Theta_{I_{2p}} \cdot \Theta_{\emptyset}^{p(g-2)} = (\Theta_{\{1\}}\cdots \Theta_{\{g\}})^p, \end{align*}$$

which is by construction a Hodge relation generated by those in degree $2$ . Multiplying $\Theta _{I_{p+1}^{\mathrm {c}}}\cdots \Theta _{I_{2p}^{\mathrm {c}}} \Theta _{\{2,\ldots ,g\}}^p$ and dividing $(2\pi i)^p$ on both sides, we get

(7.8) $$ \begin{align} \Theta_{I_1}\cdots \Theta_{I_p} \cdot (\Theta_{\emptyset}^{g-2} \Theta_{\{2,\ldots,g\}})^p = \Theta_{I_{p+1}^{\mathrm{c}}}\cdots \Theta_{I_{2p}^{\mathrm{c}}} (\Theta_{\{2\}} \cdots \Theta_{\{g\}})^p. \end{align} $$

Here we use the Hodge relation in degree $1$ : $\Theta _I \Theta _{I^{\mathrm {c}}} = 2\pi i$ for all $I \subseteq \{1,\ldots ,g\}$ .

Comparing (7.7) and (7.8), we are done if $g = 2$ , and if $g\ge 3$ it suffices to prove that

$$\begin{align*}\Theta_{\emptyset}^{g-2} \Theta_{\{2,\ldots,g\}} = \Theta_{\{2\}} \cdots \Theta_{\{g\}} \end{align*}$$

is a Hodge relation generated by those in degree $2$ . But this follows again from Lemma 7.7 applied to $I = \{2,\ldots ,g\}$ . Hence we are done.

7.5 Rational $(2,2)$ -Hodge cycles

We remark that for any $\lambda \in E^c$ and any $I,J,K,L \subseteq \{2,\ldots ,g\}$ satisfying (7.1), the vector

(7.9) $$ \begin{align} \sum_{\theta\in G} \theta(\lambda) \theta(\varepsilon_I^{(\ell_1)}\wedge \varepsilon_J^{(\ell_2)}\wedge \varepsilon_{K^{\mathrm{c}}}^{(\ell_3)}\wedge \varepsilon_{L^{\mathrm{c}}}^{(\ell_4)}) \end{align} $$

is a Hodge cycle in $B^2(A^n)$ . Moreover this Hodge cycle is not in the space generated by the algebraic cycles of codimension 1 if $\{I,J\}\not =\{K,L\}$ . It is also not known à priori that this Hodge cycle comes from an algebraic cycle. To clarify the situation let us describe the Galois orbit of a vector of the form $\varepsilon _I^{(\ell _1)}\wedge \varepsilon _J^{(\ell _2)}\wedge \varepsilon _{K^{\mathrm {c}}}^{(\ell _3)}\wedge \varepsilon _{L^{\mathrm {c}}}^{(\ell _4)}$ .

Remark 7.9. Notice that the group $E^\times $ acts naturally on $\mathrm {HC}(I,J,K,L)$ as the action of the endomorphisms E on $H^4(A^n ,{\mathbb C})$ is diagonalized in the base

$$ \begin{align*}{\mathcal B}:=\Big\{\varepsilon_{B_1}^{(\ell_1)} \wedge \varepsilon_{B_2}^{(\ell_2)} \wedge \varepsilon_{B_3}^{(\ell_3)} \wedge \varepsilon_{B_4}^{(\ell_4)} , \text{ with } B_1,B_2,B_3,B_4 \text{ subsets of } \{1,\dots,g\}\Big\} \end{align*} $$

Each $\alpha \in E^\times $ acts on $\varepsilon _{B_1}^{(\ell _1)} \wedge \varepsilon _{B_2}^{(\ell _2)} \wedge \varepsilon _{B_3}^{(\ell _3)} \wedge \varepsilon _{B_4}^{(\ell _4)}$ by multiplication by $\varepsilon _{B_1}(\alpha )\varepsilon _{B_2}(\alpha )\varepsilon _{B_3}(\alpha ) \varepsilon _{B_4}(\alpha )$ . The remark is a consequence of the fact that for any $\sigma \in G$ , any subset B of $\{1,\dots ,g\}$ and any $\alpha \in E^\times $ , we have $\sigma (\alpha \cdot \varepsilon _B^{(\ell )})=\sigma (\alpha ) \varepsilon _{\sigma (B)}^{(\ell )}$ for any $\ell \in \{1,\ldots ,n\}$ .

Here is a result on the $(2,2)$ -Hodge cycle (defined over ${\mathbb Q}$ ) on anti-Weyl CM abelian varieties.

Proposition 7.10. Assume A is anti-Weyl, i.e. $G = ({\mathbb Z}/2{\mathbb Z})^g\rtimes S_g$ . For any $I,J,K,L\subseteq \{2,\ldots ,g\}$ satisfying (7.1), the vector $\varepsilon _{IJKL}:=\varepsilon _I\wedge \varepsilon _J\wedge \varepsilon _{K^{\mathrm {c}}}\wedge \varepsilon _{L^{\mathrm {c}}}$ is in the Galois orbit of a unique vector of the form

$$ \begin{align*}\varepsilon_{\emptyset}\wedge \varepsilon_{\{2,\dots,r\}}\wedge \varepsilon_{\{2,\dots,s\}^{\mathrm{c}}}\wedge \varepsilon_{\{s+1,\dots,r\}^{\mathrm{c}}} \end{align*} $$

for an integer r such that $2\leq r\leq g$ and an integer s such that $2\leq s\leq \frac {r}{2}$ .

8.1 (Cubic) relations from the kernel

We show that there are two non-trivial cubic relations between the holomorphic periods of A which cannot be generated by algebraic relations of smaller degrees if we work only with powers of A.

Define the pairing $\langle \cdot , \cdot \rangle $ on the ${\mathbb Z}$ -module $\bigoplus _{[a]\in {\mathbb Z}/18{\mathbb Z}} {\mathbb Z}[a]$ by

$$\begin{align*}\langle [a], [b]\rangle = \begin{cases} 1 & \text{ if } [a]=[b] \\ -1 & \text{ if } [a]=[9+b] \\ 0 & \text{ otherwise}. \end{cases} \end{align*}$$

Consider the first short exact sequence in (2.7) for our situation. For the ${\mathbb Z}$ -module ${\mathbb Z}[0]\oplus {\mathbb Z}[2]\oplus {\mathbb Z}[3]\oplus {\mathbb Z}[6] \oplus {\mathbb Z}[10] \oplus {\mathbb Z}[13] \oplus {\mathbb Z}[14] \oplus {\mathbb Z}[16] \oplus {\mathbb Z}[17]$ , an element $v:=a_{[0]}[0]+a_{[2]}[2]+a_{[3]}[3]+a_{[6]}[6]+a_{[10]}[10]+a_{[13]}[13]+a_{[14]}[14]+a_{[16]}[16]+a_{[17]}[17]$ is in the kernel N if and only if $\langle v, [a]\cdot \sum _{[b] \in \Phi }[b] \rangle = 0$ for all $[a]\in {\mathbb Z}/18{\mathbb Z}$ , i.e. if and only if $\langle v, \sum _{[b]\in \Phi }[a+b] \rangle = 0$ for all $[a]\in {\mathbb Z}/18{\mathbb Z}$ , hence if and only if