2.1

Given a ring $A$ , an $A$ -algebra $A\longrightarrow B$ and an $A$ -module $M$ , we will denote by $M_{B}$ the $B$ -module $B\otimes _{A}M$ when the base ring $A$ is clear from the context. Similarly for any $A$ -scheme $X$ , we will denote by $X_{B}$ the $B$ -scheme obtained by extension of scalars to $B$ .

2.2

Let $\mathbb{A}_{f}=\mathbb{Q}\otimes _{\mathbb{Z}}\widehat{\mathbb{Z}}$ , respectively $\mathbb{A}=\mathbb{R}\times \mathbb{A}_{f}$ , denote the topological rings of finite adeles, respectively of adeles, of $\mathbb{Q}$ . The field $\mathbb{R}$ is endowed with its usual absolute value. For every prime number $p$ , we normalize the non-archimedean absolute value on $\mathbb{Q}_{p}$ by $|p|=p^{-1}$ as usual. Hence, the map $\mathbb{A}^{\times }\longrightarrow \mathbb{C}^{\times }$ defined by $(x_{v})_{v}\longmapsto \prod _{v}|x_{v}|$ induces a continuous character $|\;|:\mathbb{Q}^{\times }\backslash \mathbb{A}^{\times }\longrightarrow \mathbb{C}^{\times }$ . Every continuous character $\unicode[STIX]{x1D708}=\bigotimes _{v}^{\prime }\unicode[STIX]{x1D708}_{v}:\mathbb{Q}^{\times }\backslash \mathbb{A}^{\times }\longrightarrow \mathbb{C}^{\times }$ can be written uniquely $\unicode[STIX]{x1D708}=|\;|^{s}\unicode[STIX]{x1D708}^{0}$ for some complex number $s$ and some finite-order character $\unicode[STIX]{x1D708}^{0}$ (see [Reference BumpBum97, Proposition 3.1.2(ii)]). The infinity type of $\unicode[STIX]{x1D708}$ is by definition the complex number $s$ and the sign of $\unicode[STIX]{x1D708}$ , is by definition $\unicode[STIX]{x1D708}_{\infty }(-1)$ .

2.3

Let $I_{2}$ be the identity matrix of size two and let

The symplectic group $G=\text{GSp}(4)$ is the reductive linear algebraic group over $\mathbb{Q}$ defined as

Then $\unicode[STIX]{x1D708}:G\longrightarrow \mathbb{G}_{m}$ is a character and the derived group of $G$ is $\text{Sp}(4)=\text{Ker}\,\unicode[STIX]{x1D708}$ . We denote by $T\subset G$ the diagonal maximal torus defined as

and by $B=TU$ the standard Borel subgroup of upper triangular matrices in $G$ . We identify the group $X^{\ast }(T)$ of algebraic characters (we will also, as usual, say ‘weights’) of $T$ to the subgroup of $\mathbb{Z}^{2}\oplus \mathbb{Z}$ of triples $(k,k^{\prime },c)$ such that $k+k^{\prime }\equiv c~(\text{mod}~2)$ via

Let $\unicode[STIX]{x1D70C}_{1}=\unicode[STIX]{x1D706}(1,-1,0)$ be the short simple root and $\unicode[STIX]{x1D70C}_{2}=\unicode[STIX]{x1D706}(0,2,0)$ be the long simple root. Then the set $R\subset X^{\ast }(T)$ of roots of $T$ in $G$ is

and the subset $R^{+}\subset R$ of positive roots with respect to $B$ is

Then, the set of dominant weights is the set of $\unicode[STIX]{x1D706}(k,k^{\prime },c)$ such that $k\geqslant k^{\prime }\geqslant 0$ . For any dominant weight $\unicode[STIX]{x1D706}$ , there is an irreducible algebraic representation $V_{\unicode[STIX]{x1D706}}$ of $G$ of highest weight $\unicode[STIX]{x1D706}$ , unique up to isomorphism, and all isomorphism classes of irreducible algebraic representations of $G$ are obtained in this way. If $V_{\unicode[STIX]{x1D706}}$ is irreducible with highest weight $\unicode[STIX]{x1D706}(k,k^{\prime },c)$ , the contragredient of $V_{\unicode[STIX]{x1D706}}$ has highest weight $\unicode[STIX]{x1D706}(k,k^{\prime },-c)$ .

The Weyl group $W$ of $(G,T)$ is defined as the normalizer of $T$ in $G$ modulo its centralizer. It is a group of order eight such that the images in $W$ of the elements

generate $W$ . Then $W$ acts on $X^{\ast }(T)$ according to the rule

and we have $s_{1}.\unicode[STIX]{x1D706}(k,k^{\prime },c)=\unicode[STIX]{x1D706}(k^{\prime },k,c)$ and $s_{2}.\unicode[STIX]{x1D706}(k,k^{\prime },c)=\unicode[STIX]{x1D706}(k,-k^{\prime },c)$ which means that $s_{1}$ corresponds to the reflection associated to the short simple root $\unicode[STIX]{x1D70C}_{1}$ and $s_{2}$ to that associated to the long simple root $\unicode[STIX]{x1D70C}_{2}$ .

We shall also denote by $G^{\prime }$ the reductive linear algebraic group $\text{GL}_{2}\times _{\mathbb{G}_{m}}\text{GL}_{2}$ over $\mathbb{Q}$ , where the fiber product is over the determinant. As mentioned in the introduction, we have the embedding $\unicode[STIX]{x1D704}:G^{\prime }\longrightarrow G$ defined by

The subgroup of upper triangular matrices in $G^{\prime }$ is denoted by $B^{\prime }$ .

2.4

Let us explain our normalizations of the Haar measures on adeles groups, in the case of $G(\mathbb{A})$ . A similar discussion can be done for $G^{\prime }(\mathbb{A})$ . Consider the unitary group $\text{U}(2)=\{g\in \text{GL}(2,\mathbb{C})\mid \text{}^{t}\!\,\overline{g}g=I_{2}\}$ where $\overline{g}$ denotes the complex conjugate of $g$ . The map $\unicode[STIX]{x1D705}:\text{U}(2)\longrightarrow \text{Sp}(4,\mathbb{R})$ defined by

where $A$ and $B$ denote the real and imaginary parts of $g$ , identifies $\text{U}(2)$ with a maximal compact subgroup $K$ of $\text{Sp}(4,\mathbb{R})$ . Let $A_{G}=\mathbb{R}_{+}^{\times }$ be the identity component of the center of $G$ , write $K_{G}$ for the subgroup $A_{G}K$ of $G(\mathbb{R})$ , which is maximal compact modulo the center, and let $\unicode[STIX]{x1D709}:A_{G}\longrightarrow \mathbb{C}^{\times }$ be a continuous character. Denote again by $\unicode[STIX]{x1D709}:G(\mathbb{A})\longrightarrow \mathbb{C}^{\times }$ the extension of $\unicode[STIX]{x1D709}$ to $G(\mathbb{A})$ given by $\unicode[STIX]{x1D709}((g_{v})_{v})=|\unicode[STIX]{x1D708}(g_{\infty })|^{1/2}$ . The choice of a generator $\mathbf{1}_{G}$ of the highest exterior power of $\text{Lie}\,G_{\mathbb{R}}/\text{Lie}(A_{G}K)$ determines a left translation invariant measure on $G(\mathbb{R})/(A_{G}K)$ . Together with the Haar measure on $K$ whose total mass is one, this datum determines a left translation invariant measure $dg_{\infty }$ on $G(\mathbb{R})/A_{G}$ . For every prime number $p$ , we endow $G(\mathbb{Q}_{p})$ with the Haar measure $dg_{p}$ for which $G(\mathbb{Z}_{p})$ has volume one. Then, we have the translation invariant measure $dg=\prod _{v\leqslant \infty }dg_{v}$ on $G(\mathbb{A})$ . To introduce automorphic representations, let $L^{2}(G(\mathbb{Q})\backslash G(\mathbb{A}),\unicode[STIX]{x1D709})$ be the space of functions $f:G(\mathbb{Q})\backslash G(\mathbb{A})\longrightarrow \mathbb{C}$ such that:

1. (i) for all $z\in A_{G}$ , for all $g\in G(\mathbb{A})$ , $f(zg)=\unicode[STIX]{x1D709}(z)f(g)$ ;

2. (ii) the function $\unicode[STIX]{x1D709}^{-1}f$ is square-integrable on $A_{G}G(\mathbb{Q})\backslash G(\mathbb{A})$ .

The subspace $^{0}L^{2}(G(\mathbb{Q})\backslash G(\mathbb{A}),\unicode[STIX]{x1D709})$ of admissible cuspidal functions is a discrete sum with finite multiplicities of closed irreducible $((\mathfrak{g},K_{G})\times G(\mathbb{A}_{f}))$ -invariant subspaces, which are cuspidal automorphic representations of $G$ by definition. Here, we denote by $\mathfrak{g}$ the Lie algebra of $G(\mathbb{R})$ . Let ${\mathcal{C}}^{\infty }(G(\mathbb{Q})\backslash G(\mathbb{A}),\unicode[STIX]{x1D709})$ , respectively ${\mathcal{C}}_{c}^{\infty }(G(\mathbb{Q})\backslash G(\mathbb{A}),\unicode[STIX]{x1D709})$ , be the space of functions $f:G(\mathbb{Q})\backslash G(\mathbb{A})\longrightarrow \mathbb{C}$ such that:

1. (i) for all $z\in A_{G}$ , for all $g\in G(\mathbb{A})$ , $f(zg)=\unicode[STIX]{x1D709}(z)f(g)$ ;

2. (ii) the restriction of $f$ to $G(\mathbb{R})$ is ${\mathcal{C}}^{\infty }$ , respectively ${\mathcal{C}}^{\infty }$ and compactly supported modulo $A_{G}$ ;

3. (iii) the restriction of $f$ to $G(\mathbb{A}_{f})$ is locally constant and compactly supported.

Let ${\mathcal{C}}_{(2)}^{\infty }(G(\mathbb{Q})\backslash G(\mathbb{A}),\unicode[STIX]{x1D709})$ denote the space $L^{2}(G(\mathbb{Q})\backslash G(\mathbb{A}),\unicode[STIX]{x1D709})\cap {\mathcal{C}}^{\infty }(G(\mathbb{Q})\backslash G(\mathbb{A}),\unicode[STIX]{x1D709})$ . We have natural inclusions of $((\mathfrak{g},K_{G})\times G(\mathbb{A}_{f}))$ -modules

Finally, let ${\mathcal{C}}_{\text{cusp}}^{\infty }(G(\mathbb{Q})\backslash G(\mathbb{A}),\unicode[STIX]{x1D709})$ denote $^{0}L^{2}(G(\mathbb{Q})\backslash G(\mathbb{A}),\unicode[STIX]{x1D709})\cap {\mathcal{C}}^{\infty }(G(\mathbb{Q})\backslash G(\mathbb{A}),\unicode[STIX]{x1D709})$ . Smooth truncation to a large compact modulo the center subset induces a map

2.5

Let $\mathbb{S}=\operatorname{Res}_{\mathbb{C}/\mathbb{R}}\mathbb{G}_{m,\mathbb{C}}$ be the Deligne torus. Following Deligne and Pink, our convention for the equivalence of categories between algebraic representations of $\mathbb{S}$ in finite-dimensional $\mathbb{R}$ -vector spaces and (semisimple) mixed $\mathbb{R}$ -Hodge structures is as follows. Let $(\unicode[STIX]{x1D70C},V)$ be such a representation of $\mathbb{S}$ . Then the summand $V^{p,q}$ of $V_{\mathbb{C}}$ of type $(p,q)$ is the summand on which $\unicode[STIX]{x1D70C}(z_{1},z_{2})$ acts by multiplication by $z_{1}^{-p}z_{2}^{-q}$ for any $(z_{1},z_{2})\in \mathbb{S}(\mathbb{C})$ . In particular, any algebraic representation $V$ of $\mathbb{S}$ with central character $c$ corresponds to a pure Hodge structure of weight $-c$ . This convention disagrees with that adopted in [Reference HarrisHar04, Reference TaylorTay93] and [Reference WeissauerWei05] but agrees with that adopted in [Reference KingsKin98, Reference LemmaLem08] and [Reference PinkPin90].

3.1 Discrete series $L$ -packets

In this section and in several other places of the present article, the reader will have to be familiar with the representation theory of compact Lie groups as exposed, for example, in [Reference KnappKna86, ch. IV].

In § 2.4, we identified the unitary group $\text{U}(2)$ to a maximal compact subgroup $K$ of $\text{Sp}(4,\mathbb{R})$ via the isomorphism $\unicode[STIX]{x1D705}:\text{U}(2)\simeq K$ . Let $\mathfrak{k}$ denote the Lie algebra of $K$ and by $\mathfrak{k}_{\mathbb{C}}$ its complexification. The differential of $\unicode[STIX]{x1D705}$ induces an isomorphism of Lie algebras $d\unicode[STIX]{x1D705}:\mathfrak{gl}_{2,\mathbb{C}}\simeq \mathfrak{k}_{\mathbb{C}}$ . Let $\mathfrak{sp}_{4}$ denote the Lie algebra of $\text{Sp}(4,\mathbb{R})$ and by $\mathfrak{sp}_{4,\mathbb{C}}$ its complexification. A compact Cartan subalgebra of $\mathfrak{sp}_{4}$ is defined as $\mathfrak{h}=\mathbb{R}T_{1}\oplus \mathbb{R}T_{2}$ where

Define a $\mathbb{C}$ -basis of $\mathfrak{h}_{\mathbb{C}}^{\ast }$ by $e_{1}(T_{1})=i,e_{1}(T_{2})=0,e_{2}(T_{1})=0,e_{2}(T_{2})=i$ . The root system $\unicode[STIX]{x1D6E5}$ of the pair $(\mathfrak{sp}_{4,\mathbb{C}},\mathfrak{h}_{\mathbb{C}})$ is $\unicode[STIX]{x1D6E5}=\{\pm 2e_{1},\pm 2e_{2},\pm (e_{1}\pm e_{2})\}$ . We denote by $\unicode[STIX]{x1D6E5}_{c}$ , respectively $\unicode[STIX]{x1D6E5}_{nc}$ , the set of compact, respectively non-compact, roots in $\unicode[STIX]{x1D6E5}$ . We have $\unicode[STIX]{x1D6E5}_{c}=\{\pm (e_{1}-e_{2})\}$ and $\unicode[STIX]{x1D6E5}_{nc}=\unicode[STIX]{x1D6E5}-\unicode[STIX]{x1D6E5}_{c}$ . We choose the set of positive roots as $\unicode[STIX]{x1D6E5}^{+}=\{e_{1}-e_{2},2e_{1},e_{1}+e_{2},2e_{2}\}$ . Then, the set of compact, respectively non-compact, positive roots is $\unicode[STIX]{x1D6E5}_{c}^{+}=\unicode[STIX]{x1D6E5}_{c}\cap \unicode[STIX]{x1D6E5}^{+}$ , respectively $\unicode[STIX]{x1D6E5}_{nc}^{+}=\unicode[STIX]{x1D6E5}_{nc}\cap \unicode[STIX]{x1D6E5}^{+}$ . For each symmetric matrix $Z\in \mathfrak{gl}_{2,\mathbb{C}}$ , define the element $p_{\pm }(Z)$ of $\mathfrak{sp}_{4,\mathbb{C}}$ by

Let $X_{(\unicode[STIX]{x1D6FC}_{1},\unicode[STIX]{x1D6FC}_{2})}\in \mathfrak{sp}_{4,\mathbb{C}}$ be defined as

It follows from an easy computation that $X_{(\unicode[STIX]{x1D6FC}_{1},\unicode[STIX]{x1D6FC}_{2})}$ is a root vector corresponding to the non-compact root $(\unicode[STIX]{x1D6FC}_{1},\unicode[STIX]{x1D6FC}_{2})=\unicode[STIX]{x1D6FC}_{1}e_{1}+\unicode[STIX]{x1D6FC}_{2}e_{2}$ . If we set

we have the Cartan decomposition $\mathfrak{sp}_{4,\mathbb{C}}=\mathfrak{k}_{\mathbb{C}}\oplus \mathfrak{p}^{+}\oplus \mathfrak{p}^{-}$ .

Integral weights are defined as $(k,k^{\prime })=ke_{1}+k^{\prime }e_{2}\in \mathfrak{h}_{\mathbb{C}}^{\ast }$ with $k,k^{\prime }\in \mathbb{Z}$ and an integral weight is dominant for $\unicode[STIX]{x1D6E5}_{c}^{+}$ if $k\geqslant k^{\prime }$ . Assigning its highest weight to a finite-dimensional irreducible complex representation $\unicode[STIX]{x1D70F}$ of $K$ , we define a bijection between isomorphism classes of finite-dimensional irreducible complex representations of $K$ and dominant integral weights, whose inverse will be denoted by $(k,k^{\prime })\longmapsto \unicode[STIX]{x1D70F}_{(k,k^{\prime })}$ . Let $(k,k^{\prime })$ be a dominant integral weight and let $d=k-k^{\prime }$ . Then $\dim _{\mathbb{C}}\unicode[STIX]{x1D70F}_{(k,k^{\prime })}=d+1$ . More precisely, there exists a basis $(v_{s})_{0\leqslant s\leqslant d}$ of $\unicode[STIX]{x1D70F}_{(k,k^{\prime })}$ , such that

which we call a standard basis of $\unicode[STIX]{x1D70F}_{(k,k^{\prime })}$ . In the identities above, we agree to use the convention $v_{-1}=v_{d+1}=0$ . We will denote by $W_{K}$ the Weyl group of $(\mathfrak{k}_{\mathbb{C}},\mathfrak{h}_{\mathbb{C}})$ . According to the classification theorem [Reference KnappKna86, Theorem 9.20], we have the following result.

For the arithmetic applications we aim at, we will need the following result.

3.2 Cohomology of Siegel threefolds

Siegel threefolds are the Shimura varieties associated to the group $G$ . Let us briefly recall their definition. Let $\mathbb{S}=\operatorname{Res}_{\mathbb{C}/\mathbb{R}}\mathbb{G}_{m,\mathbb{C}}$ be the Deligne torus and let ${\mathcal{H}}$ be the $G(\mathbb{R})$ -conjugacy class of the morphism $h:\mathbb{S}\longrightarrow G_{\mathbb{R}}$ given on $\mathbb{R}$ -points by

The pair $(G,{\mathcal{H}})$ is a pure Shimura datum in the sense of [Reference PinkPin90, 2.1]. The cocharacter $\unicode[STIX]{x1D707}$ of $G_{\mathbb{C}}$ associated to the morphism $h$ as in [Reference PinkPin90, 1.3] induces

on complex points, hence it is defined over $\mathbb{Q}$ . In other terms, the reflex field of $(G,{\mathcal{H}})$ is $\mathbb{Q}$ . For any neat compact open subgroup $L$ of $G(\mathbb{A}_{f})$ , we denote by $S_{L}$ the Siegel threefold of level $L$ . This is a smooth quasi-projective $\mathbb{Q}$ -scheme such that, as complex analytic varieties, we have

where $S_{L,\mathbb{C}}^{\text{an}}$ denotes the analytification of the base change of $S_{L}$ to $\mathbb{C}$ . For $g\in G(\mathbb{A}_{f})$ and $L$ , $L^{\prime }$ two neat compact open subgroups of $G(\mathbb{A}_{f})$ such that $g^{-1}L^{\prime }g\subset L$ , right multiplication by $g$ on $S_{L,\mathbb{C}}^{\text{an}}$ descends to a morphism $[g]:S_{L^{\prime }}\longrightarrow S_{L}$ of $\mathbb{Q}$ -schemes, which is finite étale. This implies that there is an action of $G(\mathbb{A}_{f})$ on the projective system $(S_{L})_{L}$ indexed by neat compact open subgroups of $G(\mathbb{A}_{f})$ . In what follows, all compact open subgroups of $G(\mathbb{A}_{f})$ will be assumed to be neat and we will not mention it again. Because the $S_{L}$ are Shimura varieties associated to $(G,{\mathcal{H}})$ , for any algebraic representation $E$ of $G$ in a finite-dimensional $\mathbb{Q}$ -vector space, we have a polarizable variation of $\mathbb{Q}$ -Hodge structure, abusively denoted again by $E$ , on $S_{L}$ . We take the liberty not to mention the level $L$ in the notation because these variation of Hodge structures are compatible under the pull-back maps induced by the above morphisms $[g]$ . We will also denote by $E$ the local system underlying the variation of Hodge structure $E$ . This should not lead to confusion.

Let $E$ an irreducible algebraic representation of $G$ in a finite-dimensional $\mathbb{C}$ -vector space, let $\unicode[STIX]{x1D709}$ be the inverse of its central character and let $L$ be a compact open subgroup of $G(\mathbb{A}_{f})$ . Let ${\mathcal{A}}_{c}^{\ast }(S_{L},E)$ be the de Rham complex of ${\mathcal{C}}^{\infty }$ differential forms with compact support on $S_{L,\mathbb{C}}^{\text{an}}$ with values in the local system $E$ , let ${\mathcal{A}}_{(2)}^{\ast }(S_{L},E)$ be the complex of square integrable differential forms and let ${\mathcal{A}}^{\ast }(S_{L},E)$ be the complex of usual differential forms. If $\circ$ is the symbol $c,(2)$ or the empty symbol define

When $\circ$ is $c$ or $(2)$ , this definition is legitimate because the transition morphisms $[g]$ are finite étale. Moreover, these complexes carry an action of $G(\mathbb{A}_{f})$ induced by the action on $(S_{L})_{L}$ described above. In § 2.4, we introduced the Lie algebra $\mathfrak{g}$ of $G(\mathbb{R})$ and the subgroup $K_{G}=A_{G}K$ of $G(\mathbb{R})$ , which is maximal compact modulo the center. For any $(\mathfrak{g}_{\mathbb{C}},K_{G})$ -module $V$ , let $C^{\ast }(\mathfrak{g}_{\mathbb{C}},K_{G},V)$ be the $(\mathfrak{g}_{\mathbb{C}},K_{G})$ -complex of $V$ as defined in [Reference Borel and WallachBW80, I]. According to [Reference Borel and WallachBW80, VII §2], we have $G(\mathbb{A}_{f})$ -equivariant isomorphisms of complexes

which are compatible with the inclusions ${\mathcal{A}}_{c}^{\ast }(S,E)\subset {\mathcal{A}}^{\ast }(S,E)$ and

Taking cohomology, we obtain $G(\mathbb{A}_{f})$ -equivariant isomorphisms

which are compatible with the maps from cohomology with compact support to cohomology without support. Let $H_{(2)}^{\ast }(S,E)$ denote the $L^{2}$ cohomology of $S$ with coefficients in $E$ , i.e. the cohomology of the complex ${\mathcal{A}}_{(2)}^{\ast }(S,E)$ . According to [Reference BorelBor83], we have a $G(\mathbb{A}_{f})$ -equivariant isomorphism

Applying the $(\mathfrak{g}_{\mathbb{C}},K_{G})$ -cohomology functor to the maps appearing in (4) and (5) of § 2.3, we obtain the maps

where $H_{\text{cusp}}^{\ast }(S,E)$ denotes $H^{\ast }(\mathfrak{g}_{\mathbb{C}},K_{G},E\otimes _{\mathbb{C}}{\mathcal{C}}_{\text{cusp}}^{\infty }(G(\mathbb{Q})\backslash G(\mathbb{A}),\unicode[STIX]{x1D709}))$ . Let

The $((\mathfrak{g}_{\mathbb{C}},K_{G})\times G(\mathbb{A}_{f}))$ -module ${\mathcal{C}}_{\text{cusp}}^{\infty }(G(\mathbb{Q})\backslash G(\mathbb{A}),\unicode[STIX]{x1D709})$ decomposes into a direct sum

indexed by irreducible cuspidal automorphic representations of $G$ , with finite multiplicities. This induces a decomposition

into irreducible $\mathbb{C}[G(\mathbb{A}_{f})]$ -modules.

The main result of [Reference Vogan and ZuckermannVZ84] implies that the $\unicode[STIX]{x1D70B}$ contributing to the above sum, i.e. those for which $H^{3}(\mathfrak{g}_{\mathbb{C}},K_{G},\unicode[STIX]{x1D70B}_{\infty }\otimes _{\mathbb{C}}E)$ is non-zero, are those such that $\unicode[STIX]{x1D70B}_{\infty }|_{G(\mathbb{R})^{+}}\in P(E)$ (see the proof of [Reference Mokrane and TilouineMT02, Proposition 1] for more details). As a consequence, we have

where the sum is indexed by irreducible cuspidal automorphic representations of $G$ whose archimedean component belongs to $P(E)$ and where

According to [Reference Borel and WallachBW80, II. §3, Proposition 3.1], for any $\unicode[STIX]{x1D70B}_{\infty }\in P(E)$ , the $(\mathfrak{g}_{\mathbb{C}},K_{G})$ -complex of $\unicode[STIX]{x1D70B}_{\infty }\otimes _{\mathbb{C}}E$ has zero differential. Hence, we have

In this last equality, we are using the fact that the inclusion $\mathfrak{sp}_{4,\mathbb{C}}\subset \mathfrak{g}_{\mathbb{C}}$ induces an isomorphism $\mathfrak{sp}_{4,\mathbb{C}}/\mathfrak{k}_{\mathbb{C}}\simeq \mathfrak{g}_{\mathbb{C}}/(\text{Lie}K_{G})_{\mathbb{C}}$ . By the Cartan decomposition $\mathfrak{sp}_{4,\mathbb{C}}=\mathfrak{k}_{\mathbb{C}}\oplus \mathfrak{p}^{+}\oplus \mathfrak{p}^{-}$ , where $\mathfrak{p}^{\pm }$ are defined by (6), we have $\bigwedge ^{3}\mathfrak{sp}_{4,\mathbb{C}}/\mathfrak{k}_{\mathbb{C}}=\bigoplus _{p+q=3}\bigwedge ^{p}\mathfrak{p}^{+}\otimes _{\mathbb{C}}\bigwedge ^{q}\mathfrak{p}^{-}$ . As the weights for the adjoint representation of $\mathfrak{h}_{\mathbb{C}}$ on $\bigwedge ^{p}\mathfrak{p}^{+}\otimes _{\mathbb{C}}\bigwedge ^{q}\mathfrak{p}^{-}$ are the sums of $p$ distinct weights of $\mathfrak{p}^{+}$ and of $q$ distinct weights of $\mathfrak{p}^{-}$ , the reader will easily deduce the following decompositions

into irreducible $\mathbb{C}[K]$ -modules from the basic facts on irreducible representations of $K$ reviewed in § 3.1. If $E$ has highest weight $\unicode[STIX]{x1D706}(k,k^{\prime },c)$ , its infinitesimal character is $(k+2,k^{\prime }+1)$ .

From now on, let us assume that the irreducible algebraic representation $E$ takes values in a finite-dimensional $\mathbb{Q}$ -vector space. In particular, the results explained above can be applied to the complexification $E_{\mathbb{C}}$ of $E$ . For any cuspidal automorphic representation $\unicode[STIX]{x1D70B}=\unicode[STIX]{x1D70B}_{\infty }\otimes \unicode[STIX]{x1D70B}_{f}$ of $G$ such that $\unicode[STIX]{x1D70B}_{\infty }\in P(E)$ , denote again by $\unicode[STIX]{x1D70B}_{f}$ the model of $\unicode[STIX]{x1D70B}_{f}$ defined over its rationality field, which is the number field $E(\unicode[STIX]{x1D70B}_{f})$ (see Proposition 3.3). Following [Reference HarrisHar97, 2.6.2 and 2.6.3], define

and similarly

where $H_{B,!}^{3}(S,E)$ denotes the Betti cohomology with coefficients in $E$ . These are $\mathbb{Q}$ -vector spaces endowed with a $\mathbb{Q}$ -linear action of $E(\unicode[STIX]{x1D70B}_{f})$ . Extending scalars from $\mathbb{Q}$ to $\mathbb{C}$ , according to the identities (8) and (9), we have

For any $\unicode[STIX]{x1D70B}_{f}$ , the $\mathbb{C}$ -vector space

has finite dimension

according to Proposition 3.7. As a consequence, the dimension of $M_{dR}(\unicode[STIX]{x1D70B}_{f},E)$ as an $E(\unicode[STIX]{x1D70B}_{f})$ -vector space is

The same holds for $M_{B}(\unicode[STIX]{x1D70B}_{f},E)$ because of the comparison isomorphism

It follows from Saito’s formalism of mixed Hodge modules that for any open compact subgroup $L$ of $G(\mathbb{A}_{f})$ , the interior cohomology $H_{!}^{3}(S_{L},E)$ underlies a $\mathbb{Q}$ -Hodge structure. With the convention adopted here, and explained in § 2.4, this $\mathbb{Q}$ -Hodge structure is in fact pure of weight $3-c$ , where $x\longmapsto x^{c}$ is the central character of $E$ . Hence, we obtain a pure $\mathbb{Q}$ -Hodge structure on $M_{B}(\unicode[STIX]{x1D70B}_{f},E)$ :

The following definition is taken from [Reference BeilinsonBei86, §7].

Let $\text{MHS}_{A}^{+}$ denote the abelian category of real mixed $A$ -Hodge structures.

Let $\text{MHS}_{A,F}^{+}$ denote the abelian category of real mixed $A$ -Hodge structures with coefficients in $F$ .

4.1 The relative motives

In this section, we give a definition of motivic cohomology of the Shimura varieties we are interested in, with coefficients in sheaves such as $\operatorname{Sym}^{p}V_{2}\boxtimes \operatorname{Sym}^{q}V_{2}$ or $W$ . This relies on the work of Ancona [Reference AnconaAnc15] and Cisinski and Déglise [Reference Cisinski and DégliseCD09], whose ideas were initiated by Voevodsky et al. [Reference Voevodsky, Suslin and FriedlanderVSF00]. Ideally, motivic cohomology should be defined as a space of extensions in categories of mixed motivic sheaves, but these categories have not been discovered yet.

For the necessary background on relative motives of abelian schemes and relative Weil cohomologies, see [Reference AnconaAnc15, §§2 and 3], respectively. Let $A/S$ be the universal abelian surface over the Siegel threefold and let $A^{k+k^{\prime }}$ be the $(k+k^{\prime })$ th-fold fiber product over $S$ . Let $R(A^{k+k^{\prime }})$ denote the relative motive over $S$ associated to $A^{k+k^{\prime }}$ .

Let $\text{DM}_{B,c}(S)$ is the triangulated category of constructible Beilinson motives over $S$ with $\mathbb{Q}$ -coefficients as defined in [Reference Cisinski and DégliseCD09, Definition 15.1.1]. Then $W$ is an object of $\text{DM}_{B,c}(S)$ . Let $\mathbf{1}_{S}$ be the unit object of $\text{DM}_{B,c}(S)$ . Motivic cohomology with coefficients in $W$ is the $\mathbb{Q}$ -vector space defined by

The compatibility of this definition with the $K$ -theoretical one follows from [Reference Cisinski and DégliseCD09, Corollary 14.2.14]. For integers $p\geqslant 0$ and $q\geqslant 0$ , we can define $H_{{\mathcal{M}}}^{\ast }(M\times M,(\operatorname{Sym}^{p}V_{2}\,\boxtimes \,\operatorname{Sym}^{q}V_{2})(2))$ similarly. Here $M$ denotes a modular curve. Moreover, if $p,q,k$ and $k^{\prime }$ verify the conditions stated in the introduction, the relative motive $(\operatorname{Sym}^{p}V_{2}\,\boxtimes \,\operatorname{Sym}^{q}V_{2})(2)$ over $M\times M$ is naturally a direct factor of $\unicode[STIX]{x1D704}^{\ast }W$ , as Ancona’s construction is functorial. As a consequence, the motivic cohomology space $H_{{\mathcal{M}}}^{\ast }(M\times M,(\operatorname{Sym}^{p}V_{2}\boxtimes \operatorname{Sym}^{q}V_{2})(2))$ is a naturally a direct factor of $H_{{\mathcal{M}}}^{\ast }(M\times M,\unicode[STIX]{x1D704}^{\ast }W(-1))$ . As the triangulated category of constructible Beilinson motives has the formalism of Grothendieck six functors, duality [Reference Cisinski and DégliseCD09, Theorem 15.2.4] and the absolute purity isomorphism [Reference Pépin LehalleurPép15, Proposition 1.7], we have the Gysin morphism

In this setting, the definition of the regulator in Deligne–Beilinson cohomology and the compatibility with the previous one has been explained in [Reference ScholbachSch15].

4.2 Explicit description of the cohomology classes

Let us start by reviewing basic facts about Deligne–Beilinson cohomology. In what follows, we shall consider Deligne–Beilinson cohomology with coefficients in algebraic representations of the group underlying a given Shimura variety. This means that, like in [Reference KingsKin98, 2.3], we consider Deligne–Beilinson cohomology of the corresponding relative motives.

Let $\operatorname{Sch}(\mathbb{Q})$ be the category of smooth quasi-projective $\mathbb{Q}$ -schemes. Let $X$ be an object of $\operatorname{Sch}(\mathbb{Q})$ and let $n$ be an integer. For a definition of the real Deligne–Beilinson cohomology $H_{{\mathcal{D}}}^{\ast }(X/\mathbb{R},\mathbb{R}(n))$ , the reader is referred to [Reference NekovarNek94, 7]. We also have the real absolute Hodge cohomology $H_{{\mathcal{H}}}^{\ast }(X/\mathbb{R},\mathbb{R}(n))$ of $X$ with coefficients in $\mathbb{R}(n)=(2\unicode[STIX]{x1D70B}i)^{n}\mathbb{R}$ as defined in [Reference Huber and WildeshausHW98a, Definition A.2.6] and there is a canonical map

Let ${\mathcal{S}}^{m}(X/\mathbb{R},\mathbb{R}(n))$ be the vector space of ${\mathcal{C}}^{\infty }$ differential forms on $X(\mathbb{C})$ on which the map $F_{\infty }$ induced by complex conjugation on $X(\mathbb{C})$ acts by multiplication by $(-1)^{n}$ . Let ${\mathcal{S}}_{c}^{m}(X/\mathbb{R},\mathbb{R}(n))$ be the compactly supported differential forms belonging to ${\mathcal{S}}^{m}(X/\mathbb{R},\mathbb{R}(n))$ . Let us consider a smooth projective compactification $j:X\longrightarrow X^{\ast }$ and let $i:Y\longrightarrow X^{\ast }$ be the complementary reduced closed embedding. Assume that $Y$ is a normal crossing divisor and let $\unicode[STIX]{x1D6FA}_{X}^{m}\langle Y\rangle$ be the $\mathbb{C}$ -vector space of holomorphic differentials of degree $m$ on $X$ with logarithmic singularities along $Y$ , endowed with its Hodge filtration (see [Reference DeligneDel74, 3.1 and 3.2.2]). For any integer $n$ , let $\unicode[STIX]{x1D70B}_{n}:\mathbb{C}\longrightarrow \mathbb{R}(n)$ denote the map $z\longmapsto {\textstyle \frac{1}{2}}(z+(-1)^{n}\overline{z})$ .

The Eisenstein symbol [Reference BeilinsonBei88, §3] is a $\mathbb{Q}$ -linear map

where we denote by $H_{{\mathcal{M}}}^{n+1}(E^{n},\mathbb{Q}(n+1))$ the inductive limit over compact open subgroups $K$ of $\text{GL}_{2}(\mathbb{A}_{f})$ of the motivic cohomology $H_{{\mathcal{M}}}^{n+1}(E_{K}^{n},\mathbb{Q}(n+1))$ of the $n$ th-fold fiber product of the universal elliptic curve $E_{K}/M_{K}$ over the modular curve of level $K$ . The notation ${\mathcal{B}}_{n}$ stands for the space of locally constant $\mathbb{Q}$ -valued functions $\unicode[STIX]{x1D719}_{f}$ on $\text{GL}_{2}(\mathbb{A}_{f})$ such that for any $b\in \mathbb{A}_{f}$ , any $a,d\in \mathbb{Q}$ such that $ad>0$ and any $g\in \text{GL}_{2}(\mathbb{A}_{f})$ , we have

and which are invariant by right translation under $(\!\begin{smallmatrix}\widehat{\mathbb{Z}}^{\times } & \\ & 1\end{smallmatrix}\!)$ . The source of the Eisenstein symbol is indentified to a space of $\mathbb{Q}$ -valued functions ${\mathcal{F}}^{n}$ on $\text{GL}_{2}(\mathbb{A}_{f})$ in [Reference BeilinsonBei88, p. 7]. Note that the map $\unicode[STIX]{x1D713}_{f}\mapsto (\unicode[STIX]{x1D719}_{f}:g\mapsto \unicode[STIX]{x1D719}_{f}(g)=\unicode[STIX]{x1D713}_{f}(\text{det}(g)^{-1}~\text{}^{t}\!g))$ defines a $\text{GL}_{2}(\mathbb{A}_{f})$ -equivariant isomorphism ${\mathcal{F}}^{n}\simeq {\mathcal{B}}_{n}$ when $\text{GL}_{2}(\mathbb{A}_{f})$ acts on ${\mathcal{F}}^{n}$ , respectively ${\mathcal{B}}_{n}$ , by left, respectively right, translation. By definition of the motivic sheaf $\operatorname{Sym}^{n}V_{2}(1)$ , the motivic cohomology $H_{{\mathcal{M}}}^{1}(M,\operatorname{Sym}^{n}V_{2}(1))$ is a direct factor of $H_{{\mathcal{M}}}^{n+1}(E^{n},\mathbb{Q}(n+1))$ . Furthermore, the Eisenstein symbol factors through the natural inclusion $H_{{\mathcal{M}}}^{1}(M,\operatorname{Sym}^{n}V_{2}(1))\subset H_{{\mathcal{M}}}^{n+1}(E^{n},\mathbb{Q}(n+1))$ and we denote again by $\operatorname{Eis}_{{\mathcal{M}}}^{n}$ the induced map ${\mathcal{B}}_{n}\longrightarrow H_{{\mathcal{M}}}^{1}(M,\operatorname{Sym}^{n}V_{2}(1))$ . The following lemma will be very useful later.

Let us recall now the explicit description of the image of the Eisenstein symbol in Deligne–Beilinson cohomology, via the isomorphism of Proposition 4.2. Here, we follow [Reference KingsKin98, 6.3] very closely.

Let us regard the circle $\text{U}(1)$ as a maximal compact subgroup of $\text{SL}_{2}(\mathbb{R})$ in the usual way. We will also sometimes consider the maximal torus $Z_{2}(\mathbb{R})^{+}\text{U}(1)$ , where $Z_{2}(\mathbb{R})^{+}$ denotes the identity component of the center of $\text{GL}_{2}(\mathbb{R})$ . In what follows, we shall implicitly use the isomorphism between the de Rham complex on modular curves and a $(\mathfrak{gl}_{2,\mathbb{C}},Z_{2}(\mathbb{R})^{+}\text{U}(1))$ -complex which is analogous to that stated in § 3.2 for Siegel threefolds. If $\mathfrak{k}_{2}$ denotes the Lie algebra of $\text{U}(1)$ , we have the Cartan decomposition $\mathfrak{sl}_{2,\mathbb{C}}=\mathfrak{k}_{2,\mathbb{C}}\oplus \mathfrak{p}^{\prime +}\oplus \mathfrak{p}^{\prime -}$ where

Let $v^{\pm }\in \mathfrak{p}^{\prime \pm }$ denote the vector $v^{\pm }={\textstyle \frac{1}{2}}(\!\begin{smallmatrix}1 & \pm i\\ \pm i & -1\end{smallmatrix}\!)\in \mathfrak{p}^{\prime \pm }$ . Let $(X,Y)$ be a basis of the standard representation $V_{2}$ of $\text{GL}(2)$ such that $(\!\begin{smallmatrix}a & b\\ c & d\end{smallmatrix}\!)\in \text{GL}_{2}(\mathbb{Q})$ acts by

We regard $\operatorname{Sym}^{n}V_{2,\mathbb{C}}$ as the space of homogeneous polynomials of degree $n$ in the variables $X$ and $Y$ , with coefficients in $\mathbb{C}$ . For any integer $0\leqslant j\leqslant n$ , let $b_{j}^{(n)}\in \operatorname{Sym}^{n}V_{2,\mathbb{C}}$ be the vector $b_{j}^{(n)}=(iX-Y)^{j}(iX+Y)^{n-j}$ . The family $(b_{j}^{(n)})_{0\leqslant j\leqslant n}$ is a basis of $\operatorname{Sym}^{n}V_{2,\mathbb{C}}$ and we will denote by $(a_{j}^{(n)})_{0\leqslant j\leqslant n}$ the dual basis.

Let $\text{B}_{2}$ denote the standard Borel of $\text{GL}_{2}$ , and by $\text{GL}_{2}(\mathbb{R})^{+}$ , respectively $\text{B}_{2}(\mathbb{R})^{+}$ , the identity component of $\text{GL}_{2}(\mathbb{R})$ , respectively $\text{B}_{2}(\mathbb{R})$ . Given $n,c$ as above, we denote by $\unicode[STIX]{x1D706}(n,c)$ the algebraic character of the diagonal maximal torus of $\text{GL}_{2}$ defined by

For any integer $r$ such that $r\equiv n~(\text{mod}~2)$ , the function $\unicode[STIX]{x1D719}_{r}^{n}\in \operatorname{ind}_{\text{B}_{2}(\mathbb{R})^{+}}^{\text{GL}_{2}(\mathbb{R})^{+}}\unicode[STIX]{x1D706}(n+2,n)$ is defined as

where

Consider

Given $\unicode[STIX]{x1D719}_{f}\in {\mathcal{B}}_{n}$ , we can consider the vector valued series of functions

which is absolutely convergent provided $n\geqslant 1$ . Because of the limitations given by the hypothesis of Theorem 1.1, we only need to work with the Eisenstein symbol $\operatorname{Eis}_{{\mathcal{M}}}^{n}$ for $n\geqslant 1$ in this work.

For any $\unicode[STIX]{x1D719}_{f}\in {\mathcal{B}}_{n}$ , the infinite series

is absolutely convergent and defines a vector valued closed holomorphic differential one-form on $M$ , i.e. an element of

as explained in [Reference KingsKin98, 6.3]. Let $E\longrightarrow M$ denote the universal elliptic curve over $M$ . By definition of the relative motive associated to the standard representation $V_{2}$ of $\text{GL}(2)$ , the cohomology $H_{{\mathcal{D}}}^{1}(M/\mathbb{R},\operatorname{Sym}^{n}V_{2}(1))$ is a direct factor of $H_{{\mathcal{D}}}^{n+1}(E^{n}/\mathbb{R},\mathbb{R}(n+1))$ , where $E^{n}$ denotes the $n$ th-fold fiber product over $M$ .

By functoriality of the natural map between absolute Hodge and Deligne–Beilinson cohomology, the Deligne-Beilisnon cohomology classes $\operatorname{Eis}_{{\mathcal{D}}}^{p,q,W}(\unicode[STIX]{x1D719}_{f}\otimes \unicode[STIX]{x1D719}_{f}^{\prime })$ we are interested in are the image under the Gysin morphism associated to the closed embedding

of cup-products of Eisenstein classes $\operatorname{Eis}_{{\mathcal{D}}}^{p}(\unicode[STIX]{x1D719}_{f})\sqcup \operatorname{Eis}_{{\mathcal{D}}}^{q}(\unicode[STIX]{x1D719}_{f}^{\prime })$ . Hence, to get an explicit description of the classes $\operatorname{Eis}_{{\mathcal{D}}}^{p,q,W}(\unicode[STIX]{x1D719}_{f}\otimes \unicode[STIX]{x1D719}_{f}^{\prime })$ , we need an explicit description of the cup-product and of the Gysin morphism.

To give an explicit description of the Gysin morphism, we need to introduce currents and Deligne–Beilinson homology. For $X\in \operatorname{Sch}(\mathbb{Q})$ and any integer $m$ , let ${\mathcal{T}}^{\circ }(X/\mathbb{R},\mathbb{R}(m))$ be the complex of $\mathbb{R}(m)$ -valued currents on which the map $F_{\infty }$ , induced by complex conjugation on $X(\mathbb{C})$ , acts by multiplication by $(-1)^{m}$ . Let us consider as above a smooth projective compactification $j:X\longrightarrow X^{\ast }$ and let $i:Y\longrightarrow X^{\ast }$ be the complementary reduced closed embedding. Assume $Y$ is a normal crossing divisor. Let ${\mathcal{T}}_{\log }^{\circ }(X/\mathbb{R},\mathbb{C})$ be the complex of currents with logarithmic singularities along $Y$ , endowed with the Hodge filtration $(F^{r}{\mathcal{T}}_{\log }^{\circ }(X/\mathbb{R},\mathbb{C}))_{r}$ . Details on these notions are given in [Reference JannsenJan88, 1.4].

As currents are covariant for proper maps, the proposition above gives an explicit description of the Gysin morphism in Deligne–Beilinson cohomology.

Let us fix an orientation on the complex manifold $X(\mathbb{C})$ and let $\unicode[STIX]{x1D719}\in {\mathcal{S}}^{i}(X/\mathbb{R},\mathbb{R}(j))$ . Following [Reference JannsenJan88, §1], let $T_{\unicode[STIX]{x1D719}}\in {\mathcal{T}}^{i-2d_{X}}(X/\mathbb{R},\mathbb{R}(j-d_{X}))$ denote the current defined by

4.3 The use of Poincaré duality

This section explains how the Poincaré duality pairing can be used to compute the regulator. This idea is due to Beilinson [Reference BeilinsonBei88] (see also [Reference KingsKin98, 6.1]). Let us start with a general result.

The first term, respectively the second term, of the exact sequence above is obtained applying $\otimes _{\mathbb{Q}}\mathbb{R}$ to the finite-dimensional $E(\unicode[STIX]{x1D70B}_{f})$ -vector space $F^{0}M_{dR}(\unicode[STIX]{x1D70B}_{f},W)$ , respectively $M_{B}(\unicode[STIX]{x1D70B}_{f},W)^{-}(-1)$ (see Propositions 3.11 and 3.12). Let $F^{0}M_{dR}(\unicode[STIX]{x1D70B}_{f},W)^{\ast }$ be the dual of $F^{0}M_{dR}(\unicode[STIX]{x1D70B}_{f},W)$ . As a consequence, the one-dimensional $E(\unicode[STIX]{x1D70B}_{f})$ -vector space

is an $E(\unicode[STIX]{x1D70B}_{f})$ -structure of the $E(\unicode[STIX]{x1D70B}_{f})\otimes \mathbb{R}$ -module

By definition ${\mathcal{B}}(\unicode[STIX]{x1D70B}_{f},W)$ is the Beilinson $E(\unicode[STIX]{x1D70B}_{f})$ -structure, which occurs in one of the two equivalent formulations of Beilinson’s conjecture (see [Reference NekovarNek94, 6.1]). Because we are dealing with a partial $L$ -function, we do not want to use the functional equation in this work. As a consequence, we prefer to work with the Deligne $E(\unicode[STIX]{x1D70B}_{f})$ -structure.

The ranks of the $E(\unicode[STIX]{x1D70B}_{f})\otimes _{\mathbb{Q}}\mathbb{R}$ -modules of the exact sequence of Corollary 4.12 are related to multiplicities of automorphic representations in the discrete series $L$ -packet $P(W)$ .

Note that Hypothesis 4.17 and Lemma 4.15 imply that $\text{Ext}_{\text{MHS}_{\mathbb{ R}}^{+}}^{1}(\mathbb{R}(0),M_{B}(\unicode[STIX]{x1D70B}_{f},W)_{\mathbb{R}})$ is a rank-one $E(\unicode[STIX]{x1D70B}_{f})\otimes _{\mathbb{Q}}\mathbb{R}$ -module. From now on, we consider integers $p,q,k,k^{\prime }$ and a coefficient system $W$ of highest weight $\unicode[STIX]{x1D706}(k,k^{\prime },p+q+6)$ as in the statement of Theorem 1.1. Let ${\mathcal{K}}(p,q,W)$ denote the sub- $\mathbb{Q}[G(\mathbb{A}_{f})]$ -module of

generated by the image of $\operatorname{Eis}_{{\mathcal{H}}}^{p,q,W}$ and let ${\mathcal{K}}(\unicode[STIX]{x1D70B}_{f},W)$ be defined as

This is an $E(\unicode[STIX]{x1D70B}_{f})$ -submodule of $\text{Ext}_{\text{MHS}_{\mathbb{ R}}^{+}}^{1}(\mathbb{R}(0),M_{B}(\unicode[STIX]{x1D70B}_{f},W)_{\mathbb{R}})$ . We have two $E(\unicode[STIX]{x1D70B}_{f})$ -submodules of $\text{Ext}_{\text{MHS}_{\mathbb{ R}}^{+}}^{1}(\mathbb{R}(0),M_{B}(\unicode[STIX]{x1D70B}_{f},W)_{\mathbb{R}})$ ‘geometrically defined’. The elementary one, namely ${\mathcal{D}}(\unicode[STIX]{x1D70B}_{f},W)$ , is defined in terms of the de Rham and Betti cohomology. The sophisticated one, namely ${\mathcal{K}}(\unicode[STIX]{x1D70B}_{f},W)$ , is defined in terms of the regulator. Of course, whereas ${\mathcal{D}}(\unicode[STIX]{x1D70B}_{f},W)$ is non-zero by definition, we do not know at this point whether ${\mathcal{K}}(\unicode[STIX]{x1D70B}_{f},W)$ is zero or not.

Our goal is now to compute $\unicode[STIX]{x1D713}(\widetilde{v_{{\mathcal{K}}}})$ and $\unicode[STIX]{x1D713}(\widetilde{v_{{\mathcal{D}}}})$ for a well-chosen $\unicode[STIX]{x1D713}$ as above. To this end, let us recall the properties of the Poincaré duality pairing for Siegel threefolds. As the representation $W$ has highest weight $\unicode[STIX]{x1D706}(k,k^{\prime },p+q+6)$ , its contragredient has highest weight $\unicode[STIX]{x1D706}(k,k^{\prime },-p-q-6)$ . In other words, we have a perfect bilinear pairing $W\otimes W\longrightarrow \mathbb{Q}(p+q+6)$ where $\mathbb{Q}(p+q+6)$ denotes the one-dimensional $\mathbb{Q}$ -vector space on which $G$ acts by the $(p+q+6)$ th power of the multiplier character $\unicode[STIX]{x1D708}$ . This pairing induces a $G(\mathbb{A}_{f})$ -equivariant pairing

which becomes perfect after restriction to the vectors which are invariant by a compact open subgroup of $G(\mathbb{A}_{f})$ , when $\mathbb{Q}(0)$ is given the action of $G(\mathbb{A}_{f})$ by $|\unicode[STIX]{x1D708}|^{-3}$ (see [Reference TaylorTay93, p. 295]). This induces a morphism of Hodge structures

Recall that according to Proposition 3.8, we have the Hodge decomposition

where $t^{\prime }=-(k+k^{\prime }+p+q)/2$ .

4.4 From the regulator to a global integral

In this section, we shall explain how to associate a class $\unicode[STIX]{x1D714}$ whose Hodge types are as in Lemma 4.21 to some cusp forms $\unicode[STIX]{x1D6F9}$ on $G$ . When $\unicode[STIX]{x1D714}$ is associated to $\unicode[STIX]{x1D6F9}$ , we shall see that the pairing $\langle \unicode[STIX]{x1D714},\widetilde{v}_{{\mathcal{K}}}\rangle _{B}$ is an integral on $G^{\prime }(\mathbb{Q})Z^{\prime }(\mathbb{A})\backslash G^{\prime }(\mathbb{A})$ of a function of the shape $\unicode[STIX]{x1D6F9}E_{{\mathcal{H}}}$ , where $E_{{\mathcal{H}}}$ is an Eisenstein series on $G^{\prime }$ related to the Deligne–Beilinson realization $\operatorname{Eis}_{{\mathcal{D}}}^{p,q,W}(\unicode[STIX]{x1D719}_{f}\otimes \unicode[STIX]{x1D719}_{f}^{\prime })$ of the motivic classes $\operatorname{Eis}_{{\mathcal{M}}}^{p,q,W}(\unicode[STIX]{x1D719}_{f}\otimes \unicode[STIX]{x1D719}_{f}^{\prime })$ .

According to Theorem 1.1, for any $\unicode[STIX]{x1D719}_{f}\otimes \unicode[STIX]{x1D719}_{f}^{\prime }\in {\mathcal{B}}_{p}\otimes _{\mathbb{Q}}{\mathcal{B}}_{q}$ , we have the extension class

whose image in the Deligne–Beilinson cohomology $H_{{\mathcal{D}}}^{4}(S/\mathbb{R},W)$ coincides with the class of the pair of currents $(\unicode[STIX]{x1D704}_{\ast }T_{P_{{\mathcal{H}}}^{p,q}(\unicode[STIX]{x1D719}_{f}\otimes \unicode[STIX]{x1D719}_{f}^{\prime })},\unicode[STIX]{x1D704}_{\ast }T_{P_{B}^{p,q}(\unicode[STIX]{x1D719}_{f}\otimes \unicode[STIX]{x1D719}_{f}^{\prime })})$ (Proposition 4.10). Due to the above remark and Lemma 4.21, we need to compute a pairing of the shape $\langle \unicode[STIX]{x1D714},\widetilde{v}_{{\mathcal{K}}}\rangle _{B}$ where $\widetilde{v}_{{\mathcal{K}}}$ is a class in $H_{B,!}^{3}(S,W)_{\mathbb{R}}^{-}(-1)$ which is mapped to $\operatorname{Eis}_{{\mathcal{H}}}^{p,q,W}(\unicode[STIX]{x1D719}_{f}\otimes \unicode[STIX]{x1D719}_{f}^{\prime })$ by the surjection

In the following lemma, whose aim is to describe as explicitly as necessary such a lifting $\widetilde{v}_{{\mathcal{K}}}$ , we use two standard facts. The first is that Betti cohomology can be computed by closed currents. The second is the so-called Liebermann’s trick: let $p:A\longrightarrow S$ be an abelian scheme and let $W$ be a sheaf on $S$ which is a direct factor of $(R^{1}p_{\ast }\mathbb{Q}(0))^{\otimes n}(m)$ , for some integers $m$ and $n$ . Then, the cohomology $H^{i}(S,W)$ is a direct factor of $H^{i+n}(A^{n},\mathbb{Q}(t))$ , for some $t$ , where $A^{n}$ denotes the $n$ th fiber product of $A$ over $S$ . Let $r:G\longrightarrow \text{GL}(4)$ be the standard representation of $G$ . It follows from Weyl’s invariants theory that any irreducible algebraic representation $W$ of $G$ of highest weight $\unicode[STIX]{x1D706}(k,k^{\prime },c)$ is a direct factor, defined by a certain explicit Schur projector, of the representation $r^{\otimes (k+k^{\prime })}$ twisted by a power of $\unicode[STIX]{x1D708}$ (see [Reference Fulton and HarrisFH91, §17.3]). As the variation of Hodge structure associated to $r$ is $R^{1}p_{\ast }\mathbb{Q}(1)$ , the variation of Hodge structure associated to such a $W$ is a direct factor of a Tate twist of $(R^{1}\unicode[STIX]{x1D70B}_{\ast }\mathbb{Q}(0))^{\otimes (k+k^{\prime })}$ where $p:A\longrightarrow S$ is the universal abelian surface over the Siegel threefold. It is not necessary for us to be more precise. However, the reader might consult [Reference AnconaAnc15] for much more precise and general statements and their proofs.

To compute the pairing $\langle \unicode[STIX]{x1D714},[\unicode[STIX]{x1D70C}]\rangle _{B}$ , we shall use the notion of rapidly decreasing and slowly increasing differential form and the fact that the Poincaré duality pairing can be represented by a pairing between rapidly decreasing and slowly increasing differential forms. For definitions, we refer the reader to [Reference HarrisHar90, 1.3] (see also [Reference BorelBor81, 3.2]) and for the statement about the Poincaré duality pairing, we refer the reader to [Reference HarrisHar90, Proposition 1.4.4(c)].

The differential forms $\operatorname{Eis}_{B}^{p}(\unicode[STIX]{x1D719}_{f})$ and $\operatorname{Eis}_{B}^{q}(\unicode[STIX]{x1D719}_{f}^{\prime })$ are slowly increasing as explained in [Reference KingsKin98, (6.3.1)] and [Reference KingsKin98, p. 120]. Moreover, according to Proposition 4.10, we have

Hence, $P_{{\mathcal{H}}}^{p,q}(\unicode[STIX]{x1D719}_{f}\otimes \unicode[STIX]{x1D719}_{f}^{\prime })$ is slowly increasing. On the other hand, according to Proposition 3.4, the cohomology class $\unicode[STIX]{x1D714}$ can be computed by cuspidal cohomology. So, let $\unicode[STIX]{x1D6FA}$ be a cuspidal differential form representing $\unicode[STIX]{x1D714}$ . As $\unicode[STIX]{x1D6FA}$ is rapidly decreasing by definition, the differential form $\unicode[STIX]{x1D704}^{\ast }\unicode[STIX]{x1D6FA}\wedge P_{{\mathcal{H}}}^{p,q}(\unicode[STIX]{x1D719}_{f}\otimes \unicode[STIX]{x1D719}_{f}^{\prime })$ , where $\unicode[STIX]{x1D704}^{\ast }\unicode[STIX]{x1D6FA}$ denotes the restriction of $\unicode[STIX]{x1D6FA}$ to $M\times M$ , is rapidly decreasing. This differential form has values in the vector space underlying the algebraic representation

of $G^{\prime }$ . Recall that the irreducible representation $\operatorname{Sym}^{p}V_{2}\boxtimes \operatorname{Sym}^{q}V_{2}$ occurs in the isotypical decomposition of $\unicode[STIX]{x1D704}^{\ast }W(-3)$ by our choice of $p,q,k$ and $k^{\prime }$ . Hence, we have the $G^{\prime }$ -equivariant pairing

defined as the composite of the natural projection

and of the natural pairing

which is given by

Let $A_{G^{\prime }}=\mathbb{R}_{+}^{\times }$ be the identity component of the center of $G^{\prime }(\mathbb{R})$ and let $\mathfrak{g}^{\prime }$ , respectively $\mathfrak{k}^{\prime }$ , be the Lie algebra of $G^{\prime }(\mathbb{R})$ , respectively of its subgroup $A_{G^{\prime }}(\text{U}(1)\times _{\mathbb{R}^{\times }}\text{U}(1))$ . Then, with the notation of the beginning of § 4.2, we have

Let $\mathbf{1}$ be the generator of the highest exterior power $\bigwedge ^{4}\mathfrak{g}_{\mathbb{C}}^{\prime }/\mathfrak{k}_{\mathbb{C}}^{\prime }$ defined by

where $v^{\pm }={\textstyle \frac{1}{2}}(\!\begin{smallmatrix}1 & \pm i\\ \pm i & -1\end{smallmatrix}\!)\in \mathfrak{p}^{\prime \pm }$ . Evaluating the differential form $\unicode[STIX]{x1D704}^{\ast }\unicode[STIX]{x1D6FA}\wedge P_{{\mathcal{H}}}^{p,q}(\unicode[STIX]{x1D719}_{f}\otimes \unicode[STIX]{x1D719}_{f}^{\prime })$ at the tangent vector $\mathbf{1}$ , we get the rapidly decreasing vector valued function

Here ${\mathcal{C}}_{\text{rd}}^{\infty }(G^{\prime }(\mathbb{Q})\backslash G^{\prime }(\mathbb{A}))$ denotes the space of rapidly decreasing functions defined in [Reference HarrisHar90, p. 48]. Composing with the pairing defined above, finally, we get the rapidly decreasing function

Our goal is now to give an explicit formula for the integrand in the above integral. The first step is to explain precisely how to associate differential forms on $S$ to cusp forms on $G$ . Once we have the results of § 3 at our disposal, this association relies on rather elementary representation theoretic considerations.

Let $T^{\prime }$ be the maximal compact subtorus of $\text{Sp}(4,\mathbb{R})$ defined by

The Lie algebra of $T^{\prime }$ is the compact Cartan subalgebra of $\mathfrak{sp}_{4}$ that we denoted by $\mathfrak{h}$ in § 3.1. Let $A_{G}=\mathbb{R}_{+}^{\times }$ be the identity component of the center of $G(\mathbb{R})$ . For integers $n,n^{\prime },c$ such that $n+n^{\prime }\equiv c~(\text{mod}~2)$ , let $\unicode[STIX]{x1D706}^{\prime }(n,n^{\prime },c):A_{G}T^{\prime }\longrightarrow \mathbb{C}^{\times }$ denote the character defined by

and by $\unicode[STIX]{x1D706}^{\prime }(n,n^{\prime })$ the restriction of $\unicode[STIX]{x1D706}^{\prime }(n,n^{\prime },c)$ to $T^{\prime }$ . Note that the simple root $e_{1}-e_{2}$ , respectively $2e_{2}$ , defined in § 3.1, coincides with the differential at the identity matrix of the restriction to $T^{\prime }$ of the character $\unicode[STIX]{x1D706}^{\prime }(1,-1,0)$ , respectively $\unicode[STIX]{x1D706}(0,2,0)$ .

According to Lemma 4.27, to compute the integral

we need to compute pairings of the shape $\langle X_{(1,-1)}^{i}v,a_{r}^{(p)}\otimes a_{0}^{(q)}\rangle ,\langle X_{(1,-1)}^{i}v,a_{0}^{(p)}\otimes a_{s}^{(q)}\rangle$ and of the shape $\langle X_{(1,-1)}^{i}\overline{v},a_{p}^{(p)}\otimes a_{s}^{(q)}\rangle ,\langle X_{(1,-1)}^{i}v,a_{r}^{(p)}\otimes a_{q}^{(q)}\rangle$ . In fact, a lot of these pairings vanish for weight reasons.

To compute the constants $C_{1},C_{2},C_{3}$ and $C_{4}$ , let us start with a trivial remark. The family $(a_{r}^{(p)}\boxtimes a_{s}^{(q)})_{0\leqslant r\leqslant p,0\leqslant s\leqslant q}$ is a weight basis of $\operatorname{Sym}^{p}V_{2,\mathbb{C}}^{\vee }\,\boxtimes \,\operatorname{Sym}^{q}V_{2,\mathbb{C}}^{\vee }$ for the action of $\text{U}(1)^{2}$ and, by definition, the vector $v$ is a weight vector for the maximal compact torus $T^{\prime }\subset G(\mathbb{R})$ . Hence, the vector $X_{(1,-1)}^{i}v\in W(-p-q-3)$ is again a weight vector and, because $\text{U}(1)^{2}=T^{\prime }$ , its image under the $\text{U}(1)^{2}$ -equivariant projection

is equal to $\unicode[STIX]{x1D706}_{i}(v)a_{r_{i}}^{(p)}\boxtimes a_{s_{i}}^{(q)}$ for some complex number $\unicode[STIX]{x1D706}_{i}(v)$ and some integers $r_{i},s_{i}$ .

Given $i$ , is $\unicode[STIX]{x1D706}_{i}(v)$ zero or not? Note that we have the liberty to multiply the vector $w$ which enters the definition of the cohomology class $\unicode[STIX]{x1D714}$ associated to the cusp form $\unicode[STIX]{x1D6F9}$ (Lemma 4.26) by any non-zero complex number. Of course, this has the effect to multiply $\unicode[STIX]{x1D714}$ by a scalar, but does not change the ratio $\langle \unicode[STIX]{x1D714},\widetilde{v}_{{\mathcal{K}}}\rangle _{B}/\langle \unicode[STIX]{x1D714},\widetilde{v}_{{\mathcal{D}}}\rangle _{B}$ that has to be computed (see Lemma 4.21). If $\unicode[STIX]{x1D706}_{i}(v)\neq 0$ , we can replace the vector $w$ by $\unicode[STIX]{x1D706}_{i}(v)^{-1}w$ . This has the effect of replacing the vector $v=Jw$ by $\unicode[STIX]{x1D706}_{i}(v)^{-1}v$ . For this choice, $\unicode[STIX]{x1D71A}(X_{(1,-1)}^{i}v)=a_{r_{i}}^{(p)}\boxtimes a_{s_{i}}^{(q)}$ and so the computation of the pairing follows (see (14)).