1.2.1 Unitary Shimura varieties

We first replace $X_1$ and $X_2$ by closely related unitary Shimura varieties. Pick an imaginary quadratic field E that embeds in both $B_1$ and $B_2$ . Let $\mathbf {V}_1 = B_1$ and $\mathbf {V}_2 = B_2$ , viewed as (right) E-vector spaces. These are equipped with natural Hermitian forms that are of signature $(1,1)$ at the infinite place. The corresponding unitary similitude groups are given by

(1.6) $$ \begin{align} \mathrm{GU}_E(\mathbf{V}_1) \simeq (B_1^\times \times E^\times)/ F^\times, \quad \mathrm{GU}_E(\mathbf{V}_2) \simeq (B_2^\times \times E^\times)/ F^\times. \end{align} $$

Let $\mathbf {V} = \mathbf {V}_1 \oplus \mathbf {V}_2$ . Thus $\mathbf {V}$ has signature $(2,2)$ at the infinite place. Consider the maps of algebraic groups

(1.7) $$ \begin{align} B_1^\times \times B_2^\times \rightarrow \mathrm{PB}_1^\times \times \mathrm{PB}_2^\times \leftarrow \mathrm{G} (\mathrm{U}_E(\mathbf{V}_1)\times \mathrm{U}_E(\mathbf{V}_2)) /E^\times \rightarrow \mathrm{GU}_E(\mathbf{V})/E^\times. \end{align} $$

These induce maps of the associated Shimura varieties

$$\begin{align*}X_1 \times X_2 \rightarrow \tilde{X}_1 \times \tilde{X}_2 \leftarrow Y \rightarrow X \end{align*}$$

(where X is the Shimura variety associated with $\mathrm {GU}_E(\mathbf {V})/E^\times $ , etc.), which may be viewed as giving a correspondence on $(X_1 \times X_2) \times X$ . This correspondence induces a map on cohomology

$$\begin{align*}\iota^*: H^*(X) \rightarrow H^*(X_1 \times X_2). \end{align*}$$

As such, since the kernel of the map

$$\begin{align*}\mathrm{G} (\mathrm{U}_E(\mathbf{V}_1)\times \mathrm{U}_E(\mathbf{V}_2)) /E^\times \rightarrow \mathrm{PB}_1^\times \times \mathrm{PB}_2^\times \end{align*}$$

is isomorphic to

$$\begin{align*}\mathrm{G} (E^\times \times E^\times)/E^\times \simeq E^{(1)} \simeq E^\times/F^\times, \end{align*}$$

one can introduce a character $\eta $ of $E^\times /F^\times $ in the construction of the correspondence; this gives a map

$$\begin{align*}\iota_\eta^*: H^*(X) \rightarrow H^*(X_1 \times X_2). \end{align*}$$

that depends on the choice of $\eta $ .

1.2.2 Cohomological representations and Vogan–Zuckerman theory

Since the cohomology of X is given by automorphic forms [Reference Borel and Wallach10], it is natural to first look for a nontempered automorphic representation $\Pi $ of $\mathrm {GU}(\mathbf {V})$ (or say of $\mathrm {U}(\mathbf {V})$ for simplicity) which contributes to $H^2(X)$ but only to the $(1,1)$ -part. The paper of Vogan–Zuckerman [Reference Vogan and Zuckerman65] classifies cohomological representations; one finds that there is a unique nontrivial (nontempered) representation $\Pi _{\infty }^1$ of $\mathrm {U} (\mathbf {V}_{\mathbb {R}}) =\mathrm {U} (2,2)_{\mathbb {R}}$ with the property that

$$\begin{align*}H^{1,1} (\mathfrak{g}, K; \Pi_{\infty}^1) \neq 0. \end{align*}$$

The representation $\Pi _{\infty }^1$ can be realized as a cohomologically induced representation $A_{\mathfrak {q}}$ , where $\mathfrak {q}$ is a $\theta $ -stable parabolic subalgebra of $\mathfrak {g}$ with Levi component $\mathfrak {u}(1,1) \oplus \mathfrak {u}(1,1)$ . In order to construct $\Pi $ , it is first natural to look for an explicit construction of $\Pi _{\infty }^1$ which is what is accomplished in the next step.

1.2.3 An exceptional isogeny: Archimedean theta correspondence and Kudla–Millson theory

The representation $\Pi _{\infty }^1$ can be constructed as a theta lift of the trivial representation of $\mathrm {U}(1,1)$ with appropriate choices of splitting characters. However, for rather subtle reasons, this fact does not seem to be useful in our construction. Instead, we use the fact that there is an exceptional isogeny

(1.8) $$ \begin{align} \mathrm{SU}(2,2)_{\mathbb{R}} \rightarrow \mathrm{SO}(4,2)_{\mathbb{R}}. \end{align} $$

Ignoring for the moment the difference between $\mathrm {U}$ and $\mathrm {SU}$ , and between $\mathrm {O}$ and $\mathrm {SO}$ , we may view $\Pi _{\infty }^1$ as a representation of $\mathrm {O}(4,2)_{\mathbb {R}}$ , and viewed this way, the representation $\Pi _{\infty }^1$ is in fact a theta lift from $\mathrm {SL}_2$ . This fact may appear somewhat familiar to connoisseurs of Kudla–Millson theory. Indeed, Kudla–Millson theory studies certain explicit closed forms that are Poincare dual to geodesic cycles coming from embedded $\mathrm {O}(3,2)$ s in $\mathrm {O}(4,2)$ and shows that the corresponding automorphic representations of $\mathrm {O}(4,2)$ (which contribute to $H^{1,1}$ ) can be constructed as theta lifts of forms of weight $3$ on $\mathrm {SL}_2$ .

1.2.4 Inner forms

For our purposes, we need inner form versions both of the isogeny (1.8) and of the theta lift. Moreover, we need to work with similitude groups rather than isometry groups. First, the theta lift: Let B be the quaternion algebra given by $B=B_1 \cdot B_2$ in the Brauer group of $\mathbb {Q}$ . Since $B_1$ and $B_2$ are assumed to be nonisomorphic, B is a nonsplit quaternion algebra. Then there is a theta lift

where $\tilde {V}$ is a certain three-dimensional B-vector space equipped with a B-skew-Hermitian form, W is a one-dimensional B-vector space equipped with a B-Hermitian form and

denotes the space of automorphic forms on G. (To be precise, the theta lift depends on a choice of Schwartz function.) The groups $\mathrm {U}_B(W)$ and $\mathrm {U}_B(\tilde {V})$ are, respectively, the requisite inner forms of $\mathrm {SL}_2$ and $\mathrm {O}(4,2)$ . As for the isogeny, we construct (in §5) an explicit isomorphism,

(1.9) $$ \begin{align} \delta: \operatorname{PGU}_E(\mathbf{V}) \xrightarrow{\simeq} \operatorname{PGU}_B(\tilde{V})^0, \end{align} $$

which is an inner form version of equation (1.8) above for (projectivized) similitude groups.

1.2.5 The global theta lift: Schwartz forms

With this preparation, we can describe the construction of a $(1,1)$ -class on X. Let h be a modular form of weight $3$ and central character $\xi _E$ , the quadratic character associated with the extension $E/\mathbb {Q}$ , chosen such that it admits a Jacquet–Langlands transfer to $B^\times $ . Let $\tilde {\tau }_h$ be the corresponding representation of $\mathrm {GL}_2 (\mathbb {A})$ . Let $\operatorname {JL}$ denote the Jacquet–Langlands correspondence. Consider the composite maps of automorphic forms

and

We show that $\Theta \circ \operatorname {JL} (\tilde {\tau }_h)$ has trivial central character and so may be viewed as an automorphic representation of the group $\operatorname {PGU}_B(\tilde {V})^0 $ . Thus, we can consider the composite

$$\begin{align*}\Pi := \delta^{-1} \circ \Theta \circ \operatorname{JL} (\tilde{\tau}_h), \end{align*}$$

which we may view as an automorphic representation of the group $\mathrm {GU}_E (\mathbf {V}) (\mathbb {A})$ . This representation has the property that $\Pi _{\infty } \simeq \tilde {\Pi }_{\infty }^1$ , where $\tilde {\Pi }_{\infty }^1$ denotes the unique representation of $\mathrm {GU}(2,2)_{\mathbb {R}}$ with trivial central character whose restriction to $\mathrm {U}(2,2)_{\mathbb {R}}$ is isomorphic to $\Pi _{\infty }^1$ . Further, one can check that

$$\begin{align*}\dim H^{p,q} (\mathfrak{g}, K; \Pi_{\infty}) = \begin{cases} 1 & \text{if } (p,q)=(1,1) \text{ or } (3,3); \\ 2 & \text{if } (p,q)=(2,2). \end{cases} \end{align*}$$

Explicitly, we construct following the ideas of Kudla–Millson (and Funke–Millson in the higher weight case), a Schwartz form $\varphi _{\infty }$ (rather than a Schwartz function) such that with $\varphi =\varphi _{\mathrm {fin}} \otimes \varphi _{\infty }$ for any choice of a Schwartz function $\varphi _{\mathrm {fin}}$ , the theta lift

$$\begin{align*}\theta_\varphi (\phi) \end{align*}$$

may be viewed as giving a $(1,1)$ -class on X, for $\phi $ in the space of $\operatorname {JL}(\tilde {\tau }_h)$ . To be precise, the construction only depends on the restriction of $\operatorname {JL}(\tilde {\tau }_h)$ to the subgroup $\mathrm {GL}_2(\mathbb {A})^+$ (consisting of elements in $\mathrm {GL}_2(\mathbb {A})$ with positive determinant at infinity) and the vector $\phi $ must be chosen to lie in the antiholomorphic component of this restriction.

1.2.6 Nonvanishing of the restriction

Next, we show that for suitable choice of $\eta $ , h, $\varphi _{\mathrm {fin}}$ and $\phi $ , the $(1,1)$ -form $\iota _\eta ^*(\theta _\varphi (\phi ))$ is nonvanishing, when projected to the $\pi _1\boxtimes \pi _2$ -isotypic component. Let us now explain the main idea to prove this nonvanishing. Let $\omega _{f_{B_1}} $ and $\omega _{f_{B_2}}$ denote holomorphic one-forms in $H^1 (X_{B_1}, \mathbb {C})_{\pi _1}$ and $H^1 (X_{B_2}, \mathbb {C})_{\pi _2}$ , respectively. The strategy is to compute the integral

$$\begin{align*}\int_{X_{B_1} \times X_{B_2}} \iota_{\eta}^* \theta_{\varphi}(\phi) \cdot (p_1^* \omega_{f_{B_1}} \wedge \overline{p_2^* \omega_{f_{B_2}}}) \end{align*}$$

and show it is nonzero. Using the isomorphism $\delta $ from equation (1.9) and noting that the decomposition $\mathbf {V} = \mathbf {V}_1 \oplus \mathbf {V}_2$ of E-Hermitian spaces induces a decomposition $\tilde {V}=V \oplus V_0$ of B-skew-Hermitian spaces such that

$$\begin{align*}\mathrm{GU}_B(V)^0 \simeq (B_1^\times \times B_2^\times)/F^\times, \quad \mathrm{GU}_B(V_0)^0 \simeq E^\times, \end{align*}$$

and

$$\begin{align*}\delta : \mathrm{G}(\mathrm{U}_E(\mathbf{V}_1) \times \mathrm{U}_E (\mathbf{V}_2)) / E^\times \xrightarrow{\simeq} \mathrm{G}(\mathrm{U}_B(V) \times \mathrm{U}_B(V_0))^0/F^\times, \end{align*}$$

we can reduce the integral to a period on the left-hand side of the seesaw diagram below, which again involves quaternionic unitary groups:

The seesaw then implies that the period can be computed on the right where it becomes a triple product period of the form

$$\begin{align*}\int_{[\mathrm{GU}_B(W)]}\phi \cdot \overline{f_B \theta(\eta)}, \end{align*}$$

where $f_B = \theta (\overline {f_{B_1}} \boxtimes f_{B_2})$ is an automorphic form on $\mathrm {GU}_B(W) \simeq B^\times $ in the Jacquet–Langlands transfer $\pi _B$ of $\pi $ . We then show that $\eta $ , h, $\varphi _{\mathrm {fin}}$ and $\phi $ can be chosen (depending on the finite parts of $f_{B_1}$ and $f_{B_2}$ ) so as to make this triple product integral nonzero. A similar argument also shows that

$$\begin{align*}\int_{X_{B_1} \times X_{B_2}} \iota_{\eta}^* \theta_{\varphi}(\phi) \cdot (\overline{p_1^* \omega_{f_{B_1}} }\wedge p_2^* \omega_{f_{B_2}}) \end{align*}$$

is nonzero, and in fact that the induced map

$$\begin{align*}\iota_{\eta}^* \theta_{\varphi}(\phi) : H^1 (X_{B_1}, \mathbb{C}) \rightarrow H^1 (X_{B_2}, \mathbb{C}) \end{align*}$$

is an isomorphism.

1.2.7 Hodge classes

As yet we do not know that $\theta _\varphi (\phi )$ is a Hodge class. In fact, strictly speaking it is not likely to be a rational cohomology class, but we show that it lies in the $\mathbb {C}$ -span of the Hodge classes in $H^2(X)$ . The key point here is that the (expected) classification of automorphic representations implies that any automorphic representation that is nearly equivalent to $\Pi $ must have Archimedean component lying in the (unique) A-packet containing $\tilde {\Pi }_{\infty }^1$ . Moreover, this Archimedean A-packet consists of two representations $\tilde {\Pi }_{\infty }^1, \tilde {\Pi }_{\infty }^2$ and the latter contributes only to $H^4(X)$ and not $H^2(X)$ . From this, we deduce that $H^2(X,\mathbb {C})[\Pi _{\mathrm {fin}}] $ is entirely of type $(1,1)$ . (The notation $H^2(X,\mathbb {C})[\Pi _{\mathrm {fin}}] $ stands for the subspace of $H^2(X,\mathbb {C})$ on which the unramified Hecke algebra at some finite level acts by the same Hecke eigenvalues as on $\Pi _{\mathrm {fin}}$ .)

Suppose for the moment that $\Pi $ has coefficients in $\mathbb {Q}$ . Then

$$\begin{align*}H^2(X,\mathbb{Q})[\Pi_{\mathrm{fin}}] \otimes_{\mathbb{Q}} \mathbb{C} = H^2(X,\mathbb{C})[\Pi_{\mathrm{fin}}], \end{align*}$$

hence $H^2(X,\mathbb {Q})[\Pi _{\mathrm {fin}}]$ is a rational Hodge structure, pure of type $(1,1)$ . Since $\theta _\varphi (\phi )$ lies in $H^2(X,\mathbb {C})[\Pi _{\mathrm {fin}}]$ , we see that it lies in the $\mathbb {C}$ -span of $H^2(X,\mathbb {Q})[\Pi _{\mathrm {fin}}] $ and in particular is a $\mathbb {C}$ -linear combination of Hodge classes $\xi $ . We have already seen that $\theta _\varphi (\phi )\neq 0$ and moreover that its restriction to the $\pi _1 \boxtimes \pi _2$ -component of $X_1 \times X_2$ is nonzero. From this and a simple continuity argument, one deduces that there is a Hodge class $\xi \in H^2(X,\mathbb {Q})[\Pi _{\mathrm {fin}}]$ such that the induced map

$$\begin{align*}\iota_\eta ( \xi (1))^* : H^1 (X_{B_1}, \mathbb{Q})_{\pi_1} \rightarrow H^1 (X_{B_2}, \mathbb{Q})_{\pi_2} \end{align*}$$

given by

$$\begin{align*}x \mapsto p_{2,*} (p_1^* (x) \cdot \iota_{\eta}^* \xi (1)) \end{align*}$$

is an isomorphism of rational Hodge structures.

1.2.8 Galois representations

Next, we need to understand the Galois representation on $H^* (X)$ associated to the A-packet containing $\Pi $ . Again, for simplicity let us suppose $F=\mathbb {Q}$ , $d=1$ , the general case being similar. Then the expected relation between the Galois representation and the A-parameter can be deduced from Kottwitz’s conjecture (see Remark 1.4 below). In our case, we have

$$ \begin{align*} H^2(X,\mathbb{Q}_\ell)_{\Pi} &= \mathbb{Q}_\ell (-1), \\ H^4(X, \mathbb{Q}_\ell)_{\Pi} &= \mathbb{Q}_\ell(-2) \oplus \mathrm{Sym}^2(\rho_{h,\ell}) , \\ H^6 (X,\mathbb{Q}_\ell)_{\Pi} &= \mathbb{Q}_\ell (-3), \end{align*} $$

where $\rho _{h,\ell }$ is the two-dimensional $\ell $ -adic representation attached to h. From this, one deduces that as a Galois module, $H^2 (X,\mathbb {Q}_\ell )[\Pi _{\mathrm {fin}}]\simeq \mathbb {Q}_\ell (-1)^{m} $ for some integer m. For every rational prime $\ell $ , the action of $\operatorname {Gal}(\overline {\mathbb {Q}}/\mathbb {Q})$ on $\xi (1)$ is then trivial and thus $\xi (1)^*$ (viewed as acting on $\ell $ -adic cohomology via the Betti-étale comparison) is a Galois equivariant isomorphism.

1.2.9 Descending coefficients

The argument above needs a bit more care since $\Pi $ may not have coefficients in $\mathbb {Q}$ . Thus, one needs some care to ensure that the Hodge class constructed has coefficients in $\mathbb {Q}$ . This argument needed to achieve this is explained in detail in §12. Roughly, the point is to replace $H^2 (X,\mathbb {Q})[\Pi _{\mathrm {fin}}]$ by $H^2 (X,\mathbb {Q})[\mathbb {I}]$ , where $\mathbb {I}$ is the kernel of the action of the unramified Hecke algebra (with $\mathbb {Q}$ -coefficients) on $\Pi _{\mathrm {fin}}$ . Another possible source of extra coefficients is the character $\eta $ , and this needs to be handled separately.

1.2.10 The general case

This completes the outline of the proof of Theorem 1 in the case $F=\mathbb {Q}$ , $n=d=1$ , $k=2$ . The general case (assuming still that $\underline {k}=(2,\ldots ,2)$ ) is only slightly more complicated. In general, we have

$$\begin{align*}\mathrm{U}_E(\mathbf{V}) (\mathbb{R}) \simeq \mathrm{U}(2,2)^d \times \mathrm{U}(4)^{n-d}, \end{align*}$$

where the $\mathrm {U}(2,2)$ factors correspond to the places in $\Sigma $ and the $\mathrm {U}(4)$ factors to the infinite places not in $\Sigma $ . At the infinite places in $\Sigma $ , that is, where $B_1$ and $B_2$ are both split, we just imitate the constructions above. However, we need to deal as well with the infinite places where $B_1$ and $B_2$ are both ramified. At such places the representation $\Pi _{\infty }$ is trivial and the local A-packet is a singleton, consisting of just the trivial representation. This is consistent with the fact that at such places v, we have

$$\begin{align*}\mathrm{U}_E(\mathbf{V})_v\simeq \mathrm{U}(4), \quad \mathrm{U}_B(\tilde{V})_v \simeq \mathrm{O}(6), \quad \mathrm{U}_B(W)_v \simeq \mathrm{SL}_2 \end{align*}$$

and the theta lift of the weight $3$ holomorphic discrete series representation on $\mathrm {SL}_2$ is the trivial representation of $\mathrm {O}(6)$ . The conclusion then is that $H^{2d} (X,\mathbb {C})[\Pi _{\mathrm {fin}}]$ consists entirely of $(d,d)$ -classes and one can find a Hodge class $\xi \in H^{2d}(X,\mathbb {Q})[\Pi _{\mathrm {fin}}]$ such that the induced map

$$\begin{align*}\xi (d)^* : H^d (X_{B_1}, \mathbb{Q}) \rightarrow H^d (X_{B_2}, \mathbb{Q}) \end{align*}$$

given by

$$\begin{align*}x \mapsto p_{2,*} (p_1^* (x) \cdot \iota_{\eta}^* \xi (d)) \end{align*}$$

is an isomorphism of rational Hodge structures, that is also Galois invariant.

1.3.1 Local systems and normalizations

While we have stated the main result for trivial coefficients, it works equally well for local systems. In the main text, this more general case is treated.

We briefly mention the numerology in the case of general local systems. Suppose that the form $\pi $ has weights $\underline {k}=(k_1,\ldots , k_n)$ . Then (in the classical normalization) the Hodge structure of $H^*(X_i)_{\pi _i}$ is a tensor product over the places v in $\Sigma $ of a Hodge structure of type

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

Thus, $H^*(X_1)_{\pi _1} \otimes H^*(X_2)_{\pi _2}$ is a tensor product over the places v in $\Sigma $ of a Hodge structure of type

(1.10) $$ \begin{align} (2k_v-2,0) + 2 (k_v-1,k_v-1) + (0,2k_v-2). \end{align} $$

The Hodge class in $H^*(X_1)_{\pi _1} \otimes H^*(X_2)_{\pi _2}$ should come from the tensor product over the places v in $\Sigma $ of a class of type $(k_v-1,k_v-1)$ . In our construction, we pick an auxiliary form $\tilde {\tau }$ of weights $\underline {k}+{\mathbf {1}} = (k_1+1, \ldots , k_n+1)$ . Then $\operatorname {JL}( \tilde {\tau })$ corresponds to a Hodge structure which is a tensor product over the places v in $\Sigma $ of a Hodge structure of type

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

Its lift $\Pi $ to $\mathrm {U}_B(\tilde {V})$ contributes to different cohomological degrees, so there is an associated Hodge diamond which is the tensor product over the places v in $\Sigma $ of a Hodge diamond of the form

(1.11) $$ \begin{align} & \kern50pt (k_v+1,k_v+1) \\[6pt] &(2k_v,0) \quad\qquad 2 (k_v,k_v) \quad\qquad (0,2k_v) \nonumber\\[6pt] &\kern50pt (k_v-1,k_v-1).\nonumber \end{align} $$

The Hodge class in $H^*(X)_\Pi $ comes from the tensor product over the places v in $\Sigma $ of the class of type $(k_v-1,k_v-1)$ . The ‘rest’ of the Hodge structure at any place v consists of Tate twists of this $(k_v-1,k_v-1)$ class and $\mathrm {Sym}^2$ of the Hodge structure attached to $\tilde {\tau }$ .

In the main text, we use the ‘automorphic normalization’ instead of the classical normalization. This amounts to twisting the Hodge structures in equations (1.10) and (1.11) above by $(2-k_v,2-k_v)$ . This twist is therefore not visible in parallel weight $2$ .

1.3.2 The non-self-dual case

The assumption that $\pi $ has trivial central character forces $\pi $ to be self-dual and is just made for simplicity. The non-self-dual case can also be treated similarly; we do not treat this in the paper, but we outline here the main differences.

Let us denote the contragredient of $\pi $ by $\pi ^\vee $ , and let $\chi $ be the central character of $\pi $ so that $\pi \simeq \pi ^\vee \otimes \chi $ . The method described above extends to this case, except that we must choose $\eta $ such that $\eta |_{\mathbb {A}^\times _F} = \chi ^{-1} $ to compensate for the central character of $\pi $ . We remark on one unusual feature. Namely, it seems that the method outlined here naturally produces a Hodge class $\xi \in H^{2d}( X, \mathbb {Q}(\chi ^2))_\Pi $ such that

1. (i) The induced map

   $$\begin{align*}\xi(d)^*: H^d(X_{B_1}, \mathbb{Q}(\chi^{-1}))_{\pi_1^\vee} \rightarrow H^d(X_{B_2}, \mathbb{Q}(\chi))_{\pi_2} \end{align*}$$
   is an isomorphism of $\mathbb {Q}(\chi )=\mathbb {Q}(\chi ^{-1})$ -vector spaces and preserves the Hodge filtration (on tensoring with $\mathbb {C}$ ).

2. (ii) The Galois module $H^{2d}(X,\mathbb {Q}_\ell (\chi ^2))$ is isomorphic to (a sum of copies of) $\mathbb {Q}_\ell (-d) (\chi ^2)$ and the induced map

   $$\begin{align*}\xi(d) ^*: H^d(X_{B_1}, \mathbb{Q}_\ell(\chi^{-1}))_{\pi_1^\vee} \rightarrow H^d(X_{B_2}, \mathbb{Q}_\ell(\chi))_{\pi_2} \end{align*}$$
   satisfies the following Galois equivariance:
   $$\begin{align*}\sigma (\xi^*(x)) = \chi^2 (\sigma) \cdot \xi^* (\sigma (x)). \end{align*}$$
   We note that $\mathbb {Q}(\chi ^{-1})=\mathbb {Q}(\chi )$ and $\mathbb {Q}_\ell (\chi ^{-1})=\mathbb {Q}_\ell (\chi )$ .

The reason this is unusual is that one might expect to have a natural construction producing a rational Hodge class in $H^*(X_{B_1}, \mathbb {Q}(\chi ))_{\pi _1^\vee } \otimes H^*(X_{B_2}, \mathbb {Q}(\chi ))_{\pi _2}$ , since after all the Galois representation $H^*(X_{B_1}, \mathbb {Q}_\ell (\chi ))_{\pi _1^\vee } \otimes H^*(X_{B_2}, \mathbb {Q}_\ell (\chi ))_{\pi _2}$ always contains the trivial representation as a direct summand. Instead, our construction naturally produces a Hodge class (with coefficients in a number field) in $H^* (X_{B_1}, \mathbb {Q}(\chi ))_{\pi _1} \otimes H^*(X_{B_2}, \mathbb {Q}(\chi ))_{\pi _2}$ and then one has to ‘untwist’ it to produce the rational Hodge class that one expects to exist.

1.3.3 Absolute Hodge classes

The main theorem above is close to saying that the class $\xi $ is an absolute Hodge class in the sense of Deligne [Reference Deligne, Milne, Ogus and Shih15]. However, what is missing is the de Rham piece of the story, that is, in order to show that $\xi $ is absolutely Hodge, we would need to show in addition to the above that it is also de Rham rational and that for every embedding $\tau $ of $F_\Sigma $ in $\mathbb {C}$ , the class $\tau (\xi )$ is a Hodge class, whose image in $\ell $ -adic cohomology is Galois invariant for all $\ell $ . It seems difficult to show this directly. In a previous version of this paper, we expressed the hope that one might be able to deduce that $\xi $ is absolutely Hodge by showing that it satisfies a stronger property, namely that it is a motivated cycle in the sense of André. However, the strategy that we had in mind runs into a serious obstacle that we are unable to circumvent at the moment, so the problem of showing that $\xi $ is absolutely Hodge remains open. The obstacle is related to the following fact: There exist tempered L-packets $\Pi $ of representations on $\mathrm {U}_E(\mathbf {V})$ (with $\dim _E(\mathbf {V})=3$ ) which contribute to Hodge classes on the associated Shimura varieties such that the rank of the $\Pi $ -isotypic component of the space of algebraic cycles of group-theoretic origin (i.e., coming from embeddings $\mathrm {U}_E(\mathbf {V}') \hookrightarrow \mathrm {U}_E(\mathbf {V})$ , with $\dim _E(\mathbf {V}')=2$ ) is nonzero, yet is strictly smaller than the dimension of $\Pi $ -isotypic component of the space of Hodge classes. In particular, there exist Hodge classes on such varieties that are not represented by algebraic cycles coming from embedded unitary groups.

1.3.4 Functoriality for unitary groups

It would be very interesting to generalize the results of this paper to general unitary groups. The main obstacle to doing this seems to be understanding automorphic periods for the embedding

(1.12) $$ \begin{align} \mathrm{U}_E(\mathbf{V}_1) \times \mathrm{U}_E (\mathbf{V}_2) \hookrightarrow \mathrm{U}_E (\mathbf{V}), \end{align} $$

where $\dim _E(\mathbf {V}_1) = \dim _E(\mathbf {V}_2) = n$ say, with tempered representations $\pi _1$ , $\pi _2$ on $\mathrm {U}_E(\mathbf {V}_1)$ and $\mathrm {U}_E(\mathbf {V}_2)$ , respectively, and a nontempered representation $\pi $ on $\mathrm {U}_E (\mathbf {V})$ . In the case treated in this paper, this is accomplished for $n=2$ by using exceptional isogenies to relate the unitary groups above to inner forms of orthogonal groups and then using a seesaw to relate the requisite period integrals to triple product periods for $\mathrm {GL}_2$ , which are well understood and fall within the purview of the Gan–Gross–Prasad (GGP) conjectures. In the general case, these exceptional isogenies are not available. Thus, it seems important to formulate and prove analogs of the GGP conjecture in the setting of the equation (1.12) above.