Contemporary Mathematics

Introduction to plectic cohomology

J. Nekovář and A. J. Scholl

Abstract. We formulate conjectures on the existence of extra symmetries of the cohomology of Shimura varieties whose defining group is a restriction of scalars from a totally real field. We discuss evidence in its favour and potential arithmetic applications.

1. Introduction

This article^{1} is the first of a series of papers, in which we will examine the following phenomenon:

In the presence of real multiplication (by a totally real number field F of degree r), motives have a canonical and functorial additional structure (F -plectic structure).

This statement is still largely conjectural – we call it the Plectic Conjecture^{2}. More precise, but less general, versions of this conjecture will be presented in Section 6 below. In this paper we give a survey of what the conjectures are, and what kind of consequences they have. In subsequent papers we will give more precise and general formulations of the conjectures, and details of the constructions and explicit computations outlined in the later part of this paper.

The geometric objects of interest are Shimura varieties and stacks (both pure and mixed) attached to Shimura data of the form (G, 𝒳 ), where G = R_{F/ℚ}(H) is obtained by restriction of scalars from an algebraic group H defined over F, and diagrams consisting of such Shimura stacks and morphisms between them given in group-theoretical terms.

For example, the groups H appearing in the following diagram

(1.1)  \xymatrix{GL(2)_{F} ⋉ 𝔾_{a,F}^{2} \ar[d] & R_{L/F} (GL(1)_{L}) ⋉ 𝔾_{a,F}^{2} \ar[l] \ar[d] \\ GL(2)_{F} & R_{L/F} (GL(1)_{L}) \ar[l]}
\begin{matrix}c \\ ◯ \end{matrix}0000 (copyright holder)

The first author’s research was supported in part by the grant ANR-BLAN-0114.

1An expanded version of the talk given by the second author at the conference.

2The terminology comes from the Greek πλϵκτóς, meaning “twisted, wreathed” [13], and is intended to reflect the wreath product structure of the plectic Galois group , as well as the foliated structure of the complex varieties considered.

(where L is a totally imaginary quadratic extension of F) give rise to Shimura stacks

\xymatrix{𝒜 \ar[d] & 𝒜_{τ} \ar[l] \ar[d] \\ Y & \{τ\} , \ar[l]}

where Y is an open Hilbert modular variety attached to GL(2)_{F} (and a fixed level structure), τ ∈ Y is a CM point in Y and 𝒜 is the universal object over Y. Morally, 𝒜 is the quotient [Δ\A], where A is the non-existent universal Hilbert-Blumenthal abelian scheme over Y, and Δ ⊂ O_{F,+}^{×} is a subgroup (depending on the level structure) of finite index in the group of totally positive units of F. As we shall see in Section 12 below, it will be useful to consider also the stack 𝒴 = [Δ\Y], with Δ acting trivially on Y, which fits into the larger diagram

(1.2)  \xymatrix{𝒜 \ar[d] & 𝒜_{τ} \ar[l] \ar[d] \\ 𝒴 \ar[d] & [Δ\\{τ\}] \ar[l] \ar[d] \\ Y & \{τ\} . \ar[l]}

The Plectic Conjecture makes sense not only on the level of motives, but also for their various realisations. If we consider, for example, ℓ-adic étale realisations, then the notion of a plectic structure (for fixed F, which will be dropped from the notation) is straightforward. If M is a mixed motive (in the old-fashioned sense) over a field k, then its ℓ-adic realisation M_{ℓ} is a representation of the absolute Galois group Γ_{k} = Gal(k^{sep}/k) of k. This Galois group is contained in a larger plectic Galois group Γ_{k}^{plec} ⊃ Γ_{k} and a plectic structure on M_{ℓ} will be an action of Γ_{k}^{plec} extending the Galois action of Γ_{k}.

The fundamental example is that of

Γ_{ℚ} = Aut(\overline{ℚ}/ℚ) ⊂ Aut(F ⊗ \overline{ℚ}/F) =: Γ_{ℚ}^{plec} ≃ S_{r} ⋉ Γ_{F}^{r}

(the last isomorphism is non-canonical; it depends on the choice of r elements of Γ_{ℚ} extending the r embeddings of F to \overline{ℚ}).

In the best of all possible worlds one would expect the following picture. If X is a diagram of Shimura stacks for groups of the form G_{i} = R_{F/ℚ}(H_{i}) which is defined over a field k, then the structure map X → Spec(k) can be “plectified” to a cartesian diagram

(1.3)  \xymatrix{X \ar[d]^{π} \ar@{-->}[r] & X_{plec} \ar[d]^{π_{plec}} \\ Spec(k) \ar@{-->}[r]^{ι} & Spec(k)_{plec} ,}

in which the exotic “plectic” objects in the right column have the following property: if we put everywhere (pro-)étale topology, then sheaves on Spec(k) (resp. on Spec(k)_{plec}) will be Γ_{k}-modules (resp. Γ_{k}^{plec}-modules). Furthermore, the base change morphism

ι^{∗}Rπ_{plec∗}ℚ_{ℓ} ⟶ Rπ_{∗}ℚ_{ℓ}

will be an isomorphism, which means that Rπ_{plec∗}ℚ_{ℓ} will be a canonical object of the derived category of Γ_{k}^{plec}-modules with cohomology groups canonically isomorphic to the Γ_{k}-modules

H^{i}(Rπ_{∗}ℚ_{ℓ}) = H_{et}^{i}(X ⊗_{k} k^{sep}, ℚ_{ℓ}) = h^{i}(X)_{ℓ},

namely, to the ℓ-adic realisations of the cohomology motives h^{i}(X). In other words, H^{i}(Rπ_{plec∗}ℚ_{ℓ}) will define a canonical plectic structure on h^{i}(X)_{ℓ}.

Acknowledgements. The second author would like to express his thanks to the Erwin Schrödinger Institute for generous hospitality, when some of the work for this article was being done. Both authors were guests of the Fields Institute in the early stages of this project, and the present paper was completed during an extended stay at MSRI, supported by the National Science Foundation under grant no. 0932078 000.

2. Analytic cohomology of compact pure Shimura varieties

Let (G, 𝒳 ) be an arbitrary pure Shimura datum (with the minimal set of axioms [6, (2.1.1.1-3)]). Fix a point h ∈ 𝒳 and write 𝒳 = G(ℝ)/K_{∞}, where K_{∞} = G(ℝ)_{h}. Consider Y = Sh_{K}(G, 𝒳 ), for a fixed level structure^{3} K ⊂ G(ℚ̂). If the analytic space Y^{an} = ∐_{i}^{k} Γ_{i}\𝒳 is compact, then its cohomology can be written in terms of the Hilbert space decomposition

⨁_{i} L^{2}(Γ_{i}\G(ℝ)/Z(ℝ)) = ˆ⨁_{π∈Irr(G(ℝ))} π^{⊕m(π)}

using relative Lie cohomology of 𝔤 = Lie(G(ℝ))_{ℂ}:

H^{∗}(Y^{an}, ℂ) = ⨁_{π} H^{∗}(𝔤, K_{∞}; π)^{⊕m(π)},

with only finitely many π contributing.

In the plectic case G = R_{F/ℚ}(H) there are real algebraic groups H_{v} = H ⊗_{F,v} ℝ for all infinite primes v of F and product decompositions

G(ℝ) = ∏_{v|∞} H_{v}(ℝ),  K_{∞} = ∏_{v|∞} K_{∞,v},  𝒳 = ∏_{v|∞} 𝒳_{v}.

Moreover, π = ⨂_{v|∞} π_{v} with π_{v} ∈ Irr(H_{v}(ℝ)). The Künneth formula for relative Lie algebra cohomology (with 𝔥_{v} = Lie(H_{v}(ℝ))_{ℂ})

H^{∗}(𝔤, K_{∞}; π) = ⨂_{v|∞} H^{∗}(𝔥_{v}, K_{∞,v}; π_{v})

then yields a “weak Künneth decomposition” (with finitely many terms)

(2.1)  H^{∗}(Y^{an}, ℂ) = ⨁_{π}(⨂_{v|∞} H^{∗}(𝔥_{v}, K_{∞,v}; π_{v}))^{⊕m(π)}.

3Here and elsewhere, L̂ = L ⊗ ℤ̂ denotes the ring of finite adeles of a number field L.

The plectic conjecture in this particular case asserts that the appearance of ⨂_{v|∞} in (2.1) should be of a motivic origin. In particular, it should manifest itself in every cohomological realisation. In the Hodge-de Rham realisation this amounts to suitable period relations (such as those conjectured by Oda [18] and Yoshida [26]). In the real Hodge realisation, since each of the individual factors H^{∗}(𝔥_{v}, K_{∞,v}; π_{v}) has a natural Hodge decomposition induced by the Hodge torus ℂ^{∗} → H_{v}(ℝ), the total cohomology of Y has a “plectic Hodge structure”: it has a canonical ℤ^{r} × ℤ^{r}- grading

H_{ℂ}^{∗} = ⨁_{p,q∈ℤ^{r}} H^{pq}

with \overline{H^{pq}} = H^{qp} (see Section 16 below).

3. Interlude: induction and tensor induction

For any Γ_{F} -module N, the wreath product S_{r} ⋉Γ_{F}^{r} naturally acts on both N^{⊕r} and N^{⊗r}; the restrictions of these module structures to Γ_{ℚ} yield, respectively, the induced module and the tensor induction of N:

Ind_{Γ_{F}}^{Γ_{ℚ}} (N) ≃ N^{⊕r}|_{Γ_{ℚ}}, ⨂ - Ind_{Γ_{F}}^{Γ_{ℚ}} (N) ≃ N^{⊗r}|_{Γ_{ℚ}}.

A more canonical version of this wreath product is the plectic group

Γ_{ℚ}#Γ_{F} = Aut_{(Sets)-Γ_{F}} (Γ_{ℚ}),

which is canonically isomorphic to Aut(F ⊗ \overline{ℚ}/F) and which acts canonically on the intrinsically defined Ind_{Γ_{F}}^{Γ_{ℚ}} (N) and ⨂ - Ind_{Γ_{F}}^{Γ_{ℚ}} (N). The inclusion Γ_{ℚ} ⊂ Γ_{ℚ}#Γ_{F} is given by the action of Γ_{ℚ} on itself by left translations.

4. Etale cohomology of quaternionic Shimura varieties

A Shimura variety Y = Sh_{K}(G, 𝒳 ) (pure or mixed) is defined over its reflex field E ⊂ \overline{ℚ} ⊂ ℂ, which does not change if we replace (G, 𝒳 ) by the corresponding pure Shimura datum (G_{red}, 𝒳_{red}). The étale cohomology groups H^{∗} = H_{et}^{∗}(Y ⊗_{E} \overline{ℚ}, \overline{ℚ}_{ℓ}) are then ℓ-adic representations of Γ_{E}.

Consider pure quaternionic Shimura varieties: for these G = R_{F/ℚ}(H), where H = B^{×} is the multiplicative group of a quaternion division algebra B over F (so that Y is compact).

In the totally indefinite case B⊗ℝ ≃ M_{2}(ℝ)^{r} we have 𝒳 ≃ (ℂ∖ℝ)^{r}, dim(Y) = r and E = ℚ. The cohomology then decomposes as

H^{∗} = H_{int}^{∗} ⊕ H_{rest}^{∗},

with the interesting part H_{int}^{∗} = H_{int}^{r} coming from cuspidal Hilbert modular eigenforms f (of parallel weight 2) occurring only in degree r. The semi-simplification of the \overline{ℚ}_{ℓ}[Γ_{ℚ}]-module H_{int}^{r} was determined in [10, 21] (see also [14, 15]). Together with the recent proof of semi-simplicity of H_{int}^{r} [17] this yields

H_{int}^{∗} ≃ ⨁_{f} V_{f}^{⊗r}|_{Γ_{ℚ}}^{⊕m(f)},

where V_{f} is the two-dimensional ℓ-adic representation of Γ_{F} attached [19, 25, 23] to f. The remaining part of H^{∗} is isomorphic to

H_{rest}^{∗} ≃ ⨁_{χ}(χ ⊕ χ(−1))^{⊗r}|_{Γ_{ℚ}}^{⊕m(χ)},

where χ : Γ_{F} → \overline{ℚ}_{ℓ}^{×} runs through characters of finite order, and χ (resp. χ(−1) := χ ⊗ \overline{ℚ}_{ℓ}(−1)) occurs in H^{0} (resp. in H^{2}).

In particular, H^{∗} is isomorphic to a direct sum of tensor inductions of certain two-dimensional representations of Γ_{F}, and therefore has an (in general noncanonical) action of the plectic group Γ_{ℚ}#Γ_{F}. The Plectic Conjecture described in the following sections both refines and generalises this action.

In the general case we have B ⊗ ℝ ≃ M_{2}(ℝ)^{t} × ℍ^{r−t} with t ≥ 1, 𝒳 ≃ (ℂ ∖ ℝ)^{t}, dim(Y) = t and the reflex field E is determined by

Γ_{E} = Γ_{ℚ} ∩ ((S_{t} ⋉ Γ_{F}^{t}) × (S_{r−t} ⋉ Γ_{F}^{r−t})) ⊂ S_{r} ⋉ Γ_{F}^{r} .

Thanks to [10, 21, 17], the étale cohomology of Y again decomposes as H^{∗} = H_{int}^{∗} ⊕ H_{rest}^{∗}, where

H_{int}^{∗} = H_{int}^{t} ≃ ⨁_{f} V_{f}^{⊗t}|_{Γ_{E}}^{⊕m(f)}, H_{rest}^{∗} ≃ ⨁_{χ}(χ ⊕ χ(−1))^{⊗t}|_{Γ_{E}}^{⊕m(χ)},

hence is isomorphic to a direct sum of “partial” tensor inductions.

5. Plectic reflex Galois group

Let (G, 𝒳 ) be an arbitrary pure Shimura datum. Recall that, if

μ = μ_{h} : 𝔾_{m,ℂ} ⟶ G_{ℂ}

is the cocharacter attached to a point h : 𝕊 ⟶ G_{ℝ} of 𝒳 , then its conjugacy class

[μ] ∈ Hom(𝔾_{m,ℂ}, G_{ℂ})/int(G(ℂ)) = Hom(𝔾_{m,\overline{ℚ}}, G\overline{ℚ})/int(G(\overline{ℚ}))

depends only on (G, 𝒳 ). The set of conjugacy classes of cocharacters has a natural action of Γ_{ℚ}, and the stabiliser of [μ] is the absolute Galois group of the reflex field: Γ_{E} = (Γ_{ℚ})_{[μ]}.

In the plectic case G = R_{F/ℚ}(H), since G\overline{ℚ} = H⊗_{F} (F ⊗\overline{ℚ}), the set of conjugacy classes of cocharacters admits an action of the plectic Galois group Γ_{ℚ}^{plec} = Aut(F ⊗ ℚ/F) = Γ_{ℚ}#Γ_{F}, extending that of Γ_{ℚ}.

Definition 5.1. The plectic reflex Galois group of (R_{F/ℚ}(H), 𝒳 ) is the stabiliser (Γ_{ℚ}^{plec})_{[μ]} of [μ] in Γ_{ℚ}^{plec}. More generally, the plectic reflex Galois group of a mixed Shimura datum of the form (R_{F/ℚ}(H), 𝒳 ) is defined to be the plectic reflex Galois group of the corresponding pure Shimura datum (R_{F/ℚ}(H_{red}), 𝒳_{red}).

Proposition 5.2. There exists an isomorphism Γ_{ℚ}^{plec} ≃ S_{r} ⋉ Γ_{F}^{r} under which (Γ_{ℚ}^{plec})_{[μ]} corresponds to ∏_{i}(S_{r_{i}} ⋉ Γ_{F_{i}}^{r_{i}}), for suitable finite extensions F_{i}/F and r_{i} ≥ 1 such that ∑_{i} r_{i} = r.

For example, in the quaternionic case H = B^{×} the plectic reflex Galois group is isomorphic to (S_{t} ⋉ Γ_{F}^{t}) × (S_{r−t} ⋉ Γ_{F}^{r−t}) ⊂ S_{r} ⋉ Γ_{F}^{r}.

6. The ℓ-adic plectic conjecture for pure Shimura varieties

Let (G, 𝒳 ) be a pure Shimura datum for G = R_{F/ℚ}(H) with reflex field E. For every open compact subgroup K ⊂ G(ℚ̂) = H(F̂), the subgroup of central elements Z_{H}(F̂) acts on Y = Sh_{K}(G, 𝒳 ) by right multiplication, with the discrete subgroup Δ = Δ(K) := Z_{H}(F)^{+} ∩ K (which is a finitely generated abelian group) acting trivially. In the quaternionic case H = B^{×}, Δ is a subgroup of finite index in O_{F,+}^{×}.

One can attach to any algebraic representation ξ : G_{\overline{ℚ}_{ℓ}} ⟶ GL(N)_{\overline{ℚ}_{ℓ}} satisfying ξ(Δ) = 1 a lisse étale \overline{ℚ}_{ℓ}-sheaf ℒ_{ξ} on Y (if K is small enough). Its geometric étale cohomology is related to the corresponding equivariant cohomology (for the trivial action of Δ on Y)

RΓ_{et}(Y ⊗_{E} \overline{ℚ}, Δ; ℒ_{ξ}) = RΓ_{et}([Δ\Y] ⊗_{E} \overline{ℚ}, ℒ_{ξ}) ∈ D^{+}(\overline{ℚ}_{ℓ}[Γ_{E}])

by

(6.1)  RΓ_{et}([Δ\Y] ⊗_{E} \overline{ℚ}, ℒ_{ξ}) = RΓ_{et}(Y ⊗_{E} \overline{ℚ}, ℒ_{ξ}) ⊗_{ℚ} RΓ(Δ, ℚ).

In fact, one can make sense of RΓ_{et}([Δ\Y] ⊗_{E} \overline{ℚ}, ℒ_{ξ}) for arbitrary ξ (viewing ℒ_{ξ} as a sheaf on the stack 𝒴 = [Δ\Y]), but its cohomology groups vanish if ξ is irreducible and ξ(Δ) ≠ 1.

We can now state the Plectic Conjecture in the ℓ-adic setting (for Y a pure Shimura variety).

Conjecture 6.1. RΓ_{et}([Δ\Y] ⊗_{E} \overline{ℚ}, ℒ_{ξ}) has a canonical and functorial lift to an object RΓ_{et,plec}([Δ\Y] ⊗_{E} \overline{ℚ}, ℒ_{ξ}) of D^{+}(\overline{ℚ}_{ℓ}[(Γ_{ℚ}^{plec})_{[μ]}]).

Remark 6.2. In view of (6.1), this is equivalent (for ξ with ξ(Δ) = 1) to the corresponding statement for the complex RΓ_{et}(Y\overline{ℚ}, ℒ_{ξ}).

We expect (at least) functoriality with respect to Hecke correspondences and to morphisms of Shimura data (of the same type), and compatibility with products.

A weaker form of this conjecture involves only cohomology groups:

Conjecture 6.3. The Galois action of Γ_{E} = (Γ_{ℚ})_{[μ]} on étale cohomology groups H^{∗} = H_{et}^{∗}([Δ\Y] ⊗_{E} \overline{ℚ}, ℒ_{ξ}) extends to a canonical and functorial action of the plectic reflex Galois group (Γ_{ℚ}^{plec})_{[μ]}.

Proposition 6.4. Conjecture 6.3 holds for H_{et}^{0}, with the action of (Γ_{ℚ}^{plec})_{[μ]} factoring through ∏_{i} Γ_{F_{i}}^{ab}.

Proof. This follows from the explicit formula [6, 2.6] for the Galois action on π_{0}(Y ⊗_{E} \overline{ℚ}). □

Proposition 6.5. If Y is compact, then the expected expression for the Euler characteristic

∑_{k≥0}(−1)^{k}[H^{k}] ∈ G_{0}(\overline{ℚ}_{ℓ}[Γ_{E}])

lies in the image of G_{0}(\overline{ℚ}_{ℓ}[(Γ_{ℚ}^{plec})_{[μ]}]).

Sketch proof. The predicted Euler characteristic formula is a linear combination of partial tensor inductions

[ ⨂_{i} ρ_{i}^{⊗r_{i}}]|_{Γ_{E}},  ρ_{i} : Γ_{F_{i}} ⟶ GL_{n_{i}}(\overline{ℚ}_{ℓ}).  □

Proposition 6.6. In the quaternionic case H = B^{×} (including the case H = GL(2)_{F} of open Hilbert modular varieties) the Galois action of Γ_{E} extends to an action (in general, to many actions) of (Γ_{ℚ}^{plec})_{[μ]} on cohomology, commuting with all Hecke correspondences.

Proof. We can assume that the representation ξ is irreducible and that ξ(Δ) = 1; then ξ = ⨂_{v|∞} ξ_{v}, where ξ_{v} is an irreducible representation of GL(2)_{\overline{ℚ}_{ℓ}} of the

form ξ_{v} = Sym^{k_{v}−2}(\overline{ℚ}_{ℓ}^{2}) ⊗ det^{(w−k_{v})/2}, w ∈ ℤ and k_{v} ≥ 2 are integers such that k_{v} ≡ w (mod 2). In the compact case B ≠ M_{2}(F) the description of H^{∗} given in Section 4 still holds, modulo a suitable Tate twist and the fact that f will be a Hilbert modular form of weight (k_{v})_{v|∞} (moreover, H_{rest}^{∗} will be non-zero only if k_{v} = 2 for all v). In the case B = M_{2}(F) the description of H_{int}^{∗} remains the same, thanks to [3]. The Galois representation H_{rest}^{∗} in this case need not be semi-simple, because of a contribution from Eisenstein series (see [8], [5] in the case r = 2), but the corresponding extension class extends to the plectic Galois group. □

Remark 6.7. The canonical action of (Γ_{ℚ}^{plec})_{[μ]} whose existence is predicted by Conjecture 6.3 should be characterised in this case by a suitable local-global compatibility. In the case where H = B^{×} for B totally indefinite, if p is a good prime which splits completely in F, there is a factorisation of the geometric Frobenius endomorphism of the special fibre at p of the Shimura variety. The characterisation in this case should amount to the compatibility of the action of these “partial Frobenii” on ℓ-adic cohomology with the action of the corresponding elements of the plectic Galois group.

Proposition 6.8. If H = GL(2)_{F}, then Γ_{ℚ}^{plec} acts canonically on the set of cusps of the Hilbert modular variety Y and on its set of CM points.

Proof. The action on cusps is easy. The action on CM points follows from the fact that Tate’s half-transfer is plectic; cf. [16, 2.2.5], where a slightly different case is treated. □

7. Plectic ℓ-adic cohomology: the pure case

Recall that geometric and arithmetic (absolute) étale cohomology groups of a scheme of finite type X → Spec(k) are related by

RΓ_{et}(X, ℚ_{ℓ}(n)) = RΓ(Γ_{k}, RΓ_{et}(X ⊗_{k} k^{sep}, ℚ_{ℓ}(n))),

hence by a Hochschild-Serre spectral sequence

(7.1)  E_{2}^{ij} = H^{i}(Γ_{k}, H_{et}^{j}(X ⊗_{k} k^{sep}, ℚ_{ℓ}(n)) ⟹ H_{et}^{∗}(X, ℚ_{ℓ}(n)).

Let Y = Sh_{K}(G, 𝒳 ) and Δ be as in Section 6.

Definition 7.1. If Conjecture 6.1 holds and if the representation \overline{ℚ}_{ℓ}(n) (n ∈ ℤ) of Γ_{E} extends to a natural representation \overline{ℚ}_{ℓ}(n)_{plec} of (Γ_{ℚ}^{plec})_{[μ]} (for example, if H = B^{×} and t | n, then \overline{ℚ}_{ℓ}(n)_{plec} = \overline{ℚ}_{ℓ}(n/t)^{⊗t} as a representation of S_{t}⋉Γ_{F}^{t}), then the plectic étale cohomology RΓ_{et,plec}([Δ\Y], ℒ_{ξ}(n)) of [Δ\Y] with coefficients in ℒ_{ξ}(n) is defined as

RΓ((Γ_{ℚ}^{plec})_{[μ]}, RΓ_{et,plec}([Δ\Y] ⊗_{E} \overline{ℚ}, ℒ_{ξ}) ⊗ \overline{ℚ}_{ℓ}(n)_{plec}).

Likewise (when ξ(Δ) = 1) define corresponding groups with [Δ\Y] replaced by Y.

The Hochschild-Serre spectral sequence (7.1) will then be replaced by its plectic analogue

(7.2) E_{2}^{ij} = H^{i}((Γ_{ℚ}^{plec})_{[μ]}, H_{et}^{j}([Δ\Y] ⊗_{E} \overline{ℚ}, ℒ_{ξ})(n)) ⟹ H_{et,plec}^{∗}([Δ\Y], ℒ_{ξ}(n)).

In the speculative framework of (1.3), the plectic cohomology of X with coefficients in an étale sheaf ℱ on Spec(k) admitting a natural extension to a plectic étale sheaf ℱ_{plec} on Spec(k)_{plec} would simply be given by RΓ_{et}(X_{plec}, ℱ_{plec}).

8. The mixed case

The notion of a mixed Shimura datum (G, 𝒳 ) as defined in the literature [20] precludes^{4} (for G genuinely mixed, i.e. G ≠ G_{red}) the case G = R_{F/ℚ}(H) of interest here, and for a good reason: the corresponding double coset space

G(ℚ)\𝒳 × G(ℚ̂)/K

is not even an analytic space. Typically, its fibres over the corresponding pure Shimura variety are quotients of a semi-abelian variety (equipped with an O_{F} - action) by a subgroup of finite index in O_{F,+}^{×}. However there should be a good notion of the mixed Shimura stack 𝒮 h_{K}(G, 𝒳 ) in such a situation. Surprisingly, the general theory of such objects has not yet been developed. In the case H = GL(2) ⋉ 𝔾_{a}^{2}, such a stack does exist, and is just the stack 𝒜 of pointed Hilbert-Blumenthal abelian varieties (without prescribed polarisation) and level structure K. The stack 𝒴 in diagram (1.2) is the base moduli stack of HBAVs with level structure K, provided K is sufficiently small.

We then expect Conjecture 6.1 to hold (and therefore Definition 7.1 to make sense) for the stacks 𝒮 h_{K}(G, 𝒳 ) whenever G = R_{F/ℚ}(H) and, with appropriate modifications, for diagrams.

9. Motivation

Plectic cohomology (provided it exists) is interesting for the following reason. Every construction involving usual arithmetic cohomology groups of elliptic curves or modular curves (and Kuga-Sato varieties over them) can be carried out with abelian varieties with real multiplication and Hilbert modular varieties using plectic cohomology.

The output of the construction will be a usual (non-plectic) object, but in order to construct it one will have to pass through the plectic world.

10. Theta functions and classical zeta elements

Many classical zeta elements responsible for special values of L-functions of automorphic forms on GL(1) and GL(2) over ℚ (and on GL(1) over imaginary quadratic fields) arise from a motivic version of a suitably normalised two-variable theta-function Θ(z, τ) ([22, Thm. 1.2.1], [9]) or from a slightly modified function (which is a meromorphic function, rather than a section of a line bundle)_{d}Θ(z, τ) = (−1)^{(d−1)/2}Θ(z, τ)^{d^{2}}/Θ(dz, τ), where d > 1 is an integer prime to 6.

For example, the functions

g(τ) = _{d}Θ((aτ + b)/N, τ) ∈ 𝒪(Y (N))^{×} (a, b ∈ ℤ, (6N, d) = 1)

(with at least one of the integers a, b not divisible by N) are Siegel (or modular) units on the open modular curve Y (N).

4The axiom 2.1(vii) of [20], that the maximal ℝ-split central torus is ℚ-split, is violated.

Specialisations of g to CM points τ ∈ ℚ(\sqrt{−D}) (resp. to cusps) give elliptic units (resp. cyclotomic units) in abelian extensions of ℚ(\sqrt{−D}) (resp. of ℚ).

Cup products g_{1} ∪ g_{2} ∈ K_{2}(Y (N)) of two Siegel units were related by Beilinson [1] (via the regulator map) to the values of L′(f, 0) for cusp forms f of weight 2, and by Kato [9] (via an Euler system arising from norm-compatible systems of elements g_{1,n} ∪ g_{2,n} ∈ K_{2}(Y (Np^{n}))) to the Iwasawa Main Conjecture for cusp forms.

Moreover, the motivic version of _{d}Θ(z, τ) arises as the first term of the motivic elliptic polylogarithm.

11. Theta functions and cohomology

We recall the invariant cohomological definition of _{d}Θ(z, τ), for arbitrary families of elliptic curves in any of the following absolute (= arithmetic) cohomology theories H^{i}(−, j).

• Motivic cohomology H_{ℳ}^{i} (−, ℤ(j)) or H_{ℳ}^{i} (−, ℚ(j)).

• Arithmetic étale cohomology H_{et}^{i}(−, ℤ_{ℓ}(j)) or H_{et}^{i}(−, ℚ_{ℓ}(j)).

• Deligne-Beilinson absolute Hodge cohomology H_{ℋ}^{i} (−, ℝ(j)) (for varieties over ℝ or ℂ).

• (Log-)syntomic cohomology H_{syn}^{i}(−, s_{ℚ_{p}}(j)) (for varieties over a p-adic

field). Let E ⟶ Y be an elliptic curve over a scheme Y. The divisor map

(11.1)  div : H^{1}(E ∖ E[d], 1) ⟶ H^{0}(E[d], 0)^{deg=0}

admits a canonical section (characterised by a suitable compatibility with norm maps) after tensoring with ℤ[1/6d]. This gives, for each divisor D of degree zero supported on the d-torsion of E, a canonical element

θ_{D} ∈ H^{1}(E ∖ E[d], 1) ⊗ ℤ[1/6d]

with div(θ_{D}) = D. For D = d^{2}(0) − ∑_{x∈E[d]}(x) we obtain an element _{d}Θ ∈ H^{1}(E ∖ E[d], 1) (in this case there is in fact no need to tensor with ℤ[1/6d]) which is equal to the function _{d}Θ(z, τ) in the universal case when Y = Y (N) (since in motivic cohomology H_{ℳ}^{1} (X, ℤ(1)) = 𝒪(X)^{×}).

For any nowhere zero torsion section x : Y → E of order prime to d, the pullback g = x^{∗}(_{d}Θ) ∈ H^{1}(Y, 1) is a “cohomological Siegel unit”. The cup product of two such classes is an element g_{1} ∪ g_{2} ∈ H^{2}(Y, 2). In motivic cohomology, this is one of Beilinson’s elements, and its image in absolute Hodge cohomology is given by a product of two holomorphic Eisenstein series of weight two. When Y = Y (Np^{n}) for n varying, the image in p-adic étale cohomology is Kato’s Euler system.

12. Towards zeta elements over totally real fields

Our goal is to generalise the constructions in Sections 10 and 11 from ℚ to an arbitrary totally real number field F of degree r > 1. One novelty is that we will obtain formulae for leading terms of L-functions at points where they have a derivative of order r.

Instead of elliptic curves, it is natural to consider Hilbert-Blumenthal abelian varieties (= abelian varieties of dimension r equipped with an action of O_{F}). If A → S is a family of HBAV over a scheme S, the divisor map (11.1) is replaced by the residue map

H^{2r−1}(A ∖ A[d], r) ⟶ H^{0}(A[d], 0)^{deg=0},

which again admits a canonical section after tensoring with ℤ[1/d(2r + 1)!]. To simplify the notation, we assume that the ring of coefficients of our cohomology theory is a ℤ[1/d(2r + 1)!]-algebra. As in Section 11, we obtain canonical elements dΘ ∈ H^{2r−1}(A ∖ A[d], r) (d > 1) along with their pull-backs by torsion sections x : S → A of order prime to d, namely x^{∗}(_{d}Θ) ∈ H^{2r−1}(S, r).

The problem is that the element x^{∗}(_{d}Θ) is not interesting. For example, its specialisation to any \overline{ℚ}-valued point (in particular, any CM point) of S will lie in the image of the group H_{ℳ}^{2r−1}(Spec(\overline{ℚ}), ℤ(r)) ⊗ ℤ[1/d(2r + 1)!], which for r > 1 is a torsion group.

One can try to remedy the situation by considering refined cohomology theories. As in the Introduction, we should replace the universal elliptic curve E → Y (N) by the stack 𝒜 over 𝒴 = [Δ\Y], where Y is an open Hilbert modular variety attached to R_{F/ℚ}(GL(2)_{F}) and a suitable level structure, and where Δ ≃ ℤ^{r−1} (a subgroup – depending on the level structure – of O_{F,+}^{×} of finite index) acts trivially on Y. This means that we need to rerun the previous constructions for stacks (i.e. for equivariant cohomology groups).

We then obtain elements _{d}Θ ∈ H^{2r−1}(𝒜 ∖ 𝒜 [d], r) and, for any nowhere zero torsion section x : 𝒴 → 𝒜 of order prime to d,

x^{∗}(_{d}Θ) ∈ H^{2r−1}([Δ\Y], r) = ⨁_{i+j=2r−1} H^{i}(Y, r) ⊗ H^{j}(Δ, ℤ).

Taking the cap product with a fixed generator η_{Δ} ∈ H_{r−1}(Δ, ℤ) ≃ ℤ we obtain elements

g = x^{∗}(_{d}Θ) ∩ η_{Δ} ∈ H^{r}(Y, r).

The numerology is now more satisfactory, but any specialisation of g to a \overline{ℚ}-valued point of Y (in particular, any CM point) of Y will lie in the image of the torsion group

H_{ℳ}^{r} (Spec(\overline{ℚ}), ℤ(r)) ⊗ ℤ[1/d(2r + 1)!] = K_{r}^{M}(\overline{ℚ}) ⊗ ℤ[1/d(2r + 1)!].

Moreover, the cup product g_{1} ∪g_{2} ∈ H^{2r}(Y, 2r) will also be uninteresting. This can be seen in étale realisations: in the Hochschild-Serre spectral sequence

(12.1)  E_{2}^{ij} = H^{i}(Γ_{ℚ}, H_{et}^{j}(Y ⊗_{ℚ} \overline{ℚ}, ℚ_{ℓ}(2r)) ⟹ H_{et}^{∗}(Y, ℚ_{ℓ}(2r))

the groups H_{et}^{j}(Y ⊗_{ℚ} \overline{ℚ}, ℚ_{ℓ}(2r)) are interesting only for j = r (as in Section 4), but H^{i}(Γ_{ℚ}, −) = 0 for i > 2. In particular, H_{et}^{2r}(Y, ℚ_{ℓ}(2r)) is uninteresting if r > 2. If r = 2, the relevant Galois H^{2} may be non-zero, but is not expected to contain any non-trivial images of elements of motivic cohomology (since motivic H^{2} over number fields is expected to vanish).

To sum up, the elements g ∈ H^{r}(Y, r) are no good, either. We need to do something else.

13. Plectic theta elements

The solution is to plectify the previous construction. In order to do that, we will assume in Sections 13–15 that Conjecture 6.1 holds for all objects appearing in the diagram (1.2) (see Section 6 for partial results in this direction), as well as for complements of torsion sections arising from the level structure. The plectic formalism from Section 7 (with (Γ_{ℚ}^{plec})_{[μ]} = Γ_{ℚ}^{plec} = Aut(F ⊗ \overline{ℚ}/F) ≃ S_{r} ⋉ Γ_{F}^{r} for Y, 𝒴 = [Δ\Y] and 𝒜) then applies and we obtain plectic theta elements (more precisely, their ℓ-adic realisations)

dΘ_{plec} ∈ H_{et,plec}^{2r−1}(𝒜 ∖ 𝒜 [d], ℚ_{ℓ}(r)),

their pull-backs by torsion sections arising from the level structure

x^{∗}(_{d}Θ_{plec}) ∈ H_{et,plec}^{2r−1}([Δ\Y], ℚ_{ℓ}(r))

and plectic Siegel classes

g_{plec} = x^{∗}(_{d}Θ) ∩ η_{Δ} ∈ H_{et,plec}^{r}(Y, ℚ_{ℓ}(r)).

14. Specialisations of plectic Siegel classes and Stark’s conjectures

The plectic reflex group of a point τ ∈ Y which has complex multiplication by a quadratic extension L of F is isomorphic to S_{r} ⋉ Γ_{L}^{r}. The specialisation g_{plec}(τ) of g_{plec} at τ will be contained in the group

H_{et,plec}^{r}({τ}, ℚ_{ℓ}(r)) ≃ H^{r}(S_{r} ⋉ Γ_{L}^{r}, ℚ_{ℓ}(r) ⊗ ℤ[Gal(L_{τ} /L)])
≃ ⋀_{ℚ_{ℓ}[Gal(L_{τ} /L)]}^{r}H^{1}(Γ_{L_{τ}} , ℚ_{ℓ}(1)),

where L_{τ} /L is a finite abelian extension depending on τ, and where the isomorphisms come from the plectic Hochschild-Serre spectral sequence (7.2) and Shapiro’s Lemma.

If the above construction has a motivic version, then g_{plec}(τ) will be the image of a motivic element

g_{ℳ ,plec} (τ) ∈ ⋀_{ℚ[Gal(L_{τ} /L)]}^{r}H_{ℳ}^{1} (L_{τ} , ℚ(1)) ≃ ⋀_{ℚ[Gal(L_{τ} /L)]}^{r}(L_{τ}^{×} ⊗ ℚ),

whose existence is predicted by a variant of Stark’s conjectures.

Similarly, specialisations of g_{plec} to cusps will yield elements of

⋀_{ℚ_{ℓ}[Gal(F′/F)]}^{r}H^{1}(Γ_{F′}, ℚ_{ℓ}(1)),

whose motivic versions will be contained in ⋀_{ℚ[Gal(F′/F)]}^{r}(F^{′×}⊗ℚ), for finite abelian extensions F′/F.

15. Plectic constructions involving plectic Siegel classes

If g_{plec,1}, g_{plec,2} ∈ H_{et,plec}^{r}(Y, ℚ_{ℓ}(r)) are two plectic Siegel classes, we can consider their cup product

g_{plec,1} ∪ g_{plec,2} ∈ H_{et,plec}^{2r}(Y, ℚ_{ℓ}(2r)).

The plectic Hochschild-Serre spectral sequence (7.2) gives a map

H_{et,plec}^{2r}(Y, ℚ_{ℓ}(2r)) ⟶ E_{2}^{rr} = H^{r}(S_{r} ⋉ Γ_{F}^{r} , H_{et}^{r}(Y ⊗_{ℚ} \overline{ℚ}, ℚ_{ℓ})(2r)).

Geometric étale cohomology H_{et}^{r}(Y ⊗_{ℚ} \overline{ℚ}, \overline{ℚ}_{ℓ}) contains as direct summands representations of S_{r} ⋉ Γ_{F}^{r} of the form

V_{f}^{⊗r} ⊗ ℤ[Gal(F′/F)],

where f is a cuspidal Hilbert eigenform of parallel weight 2 over F, F′/F is a finite abelian extension (depending on the level structure) and S_{r} (resp. Γ_{F}^{r}) acts on ℤ[Gal(F′/F)] trivially (resp. through the product map Γ_{F}^{r} → Γ_{F}^{ab}). Projecting the image of g_{plec,1} ∪ g_{plec,2} on such a summand and using Shapiro’s Lemma, we obtain an element

(g_{plec,1} ∪ g_{plec,2})_{f} ∈ H^{r}(S_{r} ⋉ Γ_{F}^{r} , (V_{f}(2))^{⊗r} ⊗ ℤ[Gal(F′/F)]) ≃ ⋀_{ℚ_{ℓ}[Gal(F′/F)]}^{r} H^{1}(Γ_{F′}, V_{f}(2)).

Such elements (with ℤ_{ℓ}-coefficients) form an Euler system of rank r, which generalises Kato’s Euler system.

Fix an embedding i : Y ↪ Y_{1} × Y_{2} given by group-theoretical data, where Y_{1} and Y_{2} are also open Hilbert modular varieties (attached to the same field F). Let f_{1}, f_{2} be cuspidal Hilbert eigenforms of parallel weight 2 over F occurring in the geometric cohomology of Y_{1} and Y_{2}, respectively.

If we apply to g_{plec} the plectic Gysin map

i_{∗} : H_{et,plec}^{r}(Y, ℚ_{ℓ}(r)) ⟶ H_{et,plec}^{3r}(Y_{1} × Y_{2}, ℚ_{ℓ}(2r))

composed with the map

H_{et,plec}^{3r}(Y_{1} × Y_{2}, \overline{ℚ}_{ℓ}(2r)) ⟶ H^{r}(S_{r} ⋉ Γ_{F}^{r} , (V_{f_{1}} ⊗ V_{f_{2}})(2))^{⊗r} ⊗ ℤ[Gal(F′/F)])

coming from the plectic Hochschild-Serre spectral sequence, we obtain an element of

⋀_{ℚ_{ℓ}[Gal(F′/F)]}^{r} H^{1}(Γ_{F′}, V_{f_{1}} ⊗ V_{f_{2}}(2)),

for a certain finite abelian extension F′/F. This is a generalisation of [11].

One can also replace Y_{1} × Y_{2} by a Hilbert modular variety attached to a totally real quadratic extension F_{0} of F. In this case V_{f_{1}} ⊗ V_{f_{2}} is replaced by the tensor induction of V_{f_{0}}, for a cuspidal eigenform f_{0} of weight 2 over F_{0}, yielding an element of

⋀_{ℚ_{ℓ}[Gal(F′/F)]}^{r} H^{1}(Γ_{F′}, (⨂ - Ind_{Γ_{F_{0}}}^{Γ_{F}} (V_{f_{0}}))(2)).

16. Plectic Hodge theory

The constructions of the previous two sections are for the moment conjectural, relying on the existence of the plectic structure on étale cohomology. However in the analogous setting of real Hodge structures, it is possible to give a fairly complete unconditional description of the plectic structure, and to construct a Hodge-theoretic version of the plectic theta elements of Section 13. Explicit computations then lead to formulae for special values of L-functions, as predicted by Stark’s conjectures (for L-functions of abelian characters of CM-fields) and Beilinson’s conjectures (for L-functions of Hilbert modular forms). In this and the following section we will give some indications as to how this works. Further details, including the foundations of plectic Hodge theory, will appear in a later paper in this series.

For cohomology with real coefficients, all that matters is the algebra F ⊗ ℝ, so for the moment we will largely ignore F, and simply fix an integer r ≥ 1. For n = (n_{1}, …, n_{r}) ∈ ℤ^{r} we shall write |n| = ∑ n_{i}.

Definition 16.1. A (pure) plectic real Hodge structure of weight n ∈ ℤ is a finite-dimensional real vector space V whose complexification carries a grading indexed by ℤ^{2r}:

V_{ℂ} := V ⊗_{ℝ} ℂ = ⨁_{p,q∈ℤ^{r}} V^{pq}

such that \overline{V^{pq}} = V^{qp}, and V^{pq} = 0 unless |p + q| = n.

Such things form the objects of weight n of an obvious Tannakian category, equivalent to the category of representations of the real algebraic group 𝕊^{r}, where 𝕊 = R_{ℂ/ℝ}(𝔾_{m,ℂ}) is the Deligne torus. Restricting to the diagonal 𝕊 ⊂ 𝕊^{r} gives a forgetful functor from plectic to usual Hodge structures. One has the “plectic Tate object” ℝ(r)_{plec}, for which the only non-vanishing V^{pq} occurs for p = q = (−1, …, −1), and whose image under the forgetful functor is the usual Tate structure ℝ(r).

Given a plectic Hodge structure V one can define r “partial Hodge filtrations” (F_{j}^{•})_{1≤j≤r} on V_{ℂ}, given by

F_{j}^{p}V_{ℂ} = ∑_{\begin{matrix}p,q∈ℤ^{r} \\ p_{j}≥p \end{matrix}} V^{pq}.

The discussion of Section 2 shows that the real cohomology of a compact pure Shimura variety Y associated to a group of the form R_{F/ℚ}(H) has a canonical plectic Hodge structure.

When Y is not compact, we need the notion of a plectic mixed Hodge structure. Recall [7] that the category ℝMHS of real mixed Hodge structures is equivalent to the category of representations of a real pro-algebraic group 𝒢_{ℋ} , which is the semidirect product of 𝕊 and a certain pro-unipotent group 𝒰 , whose Lie algebra has an explicit description, as follows. A representation V of 𝒢_{ℋ} gives rise to two bigradings V_{ℂ} = ⨁ V_{F}^{pq} = ⨁ V pq ; \overline{F}, the first of which comes from the action of 𝕊 ⊂ 𝒢_{ℋ} , together with a nilpotent endomorphism δ ∈ End(V_{ℂ}) such that V pq ; \overline{F} =

exp(δ)(V_{F}^{pq}). It satisfies δ(V_{F}^{pq}) ⊂ ∑_{p′<p,q′<q} V_{F}^{p′q′}, and the graded components of δ are the images of the (free) generators of Lie 𝒰_{ℂ}. The filtrations F, \overline{F} and W can be recovered from the bigradings as

F^{p}V_{ℂ} = ∑_{p′≥p} V_{F}^{p′q}, \overline{F}^{q}V_{ℂ} = ∑_{q′≥q} V pq′ ; \overline{F}, W_{n}V_{ℂ} = ∑_{p+q≤n} V_{F}^{pq} = ∑_{p+q≤n} V pq ; \overline{F}.

We may then simply define the category ℝMHS_{plec} of plectic real mixed Hodge structures to be the category Rep_{ℝ}(𝒢_{ℋ}^{r} ).

If V is an object of ℝMHS_{plec} then from the action of 𝕊^{r} ⊂ 𝒢_{ℋ}^{r} it acquires a ℤ^{2r}-grading, and therefore a weight filtration W_{•} (defined over ℝ) and r partial Hodge filtrations F_{j}^{•} (defined on V_{ℂ}) in such a way that gr_{n}^{W} V becomes a pure plectic Hodge structure of weight n. Conversely, suppose one is given a real vector space V together with a real filtration W_{•} on V and r complex filtrations F_{j}^{•}, such that on each gr_{n}^{W} V the F_{j}^{•} induce a pure plectic Hodge structure. Then, provided the filtrations satisfy a further technical compatibility which we will not write down here, V is a plectic mixed Hodge structure.

It is natural to expect that the real cohomology of a Shimura variety/stack (pure or mixed), whose associated group is a restriction of scalars from F, carries a natural plectic mixed Hodge structure. It seems hard to see this directly from the Lie algebra-theoretic description of cohomology in the non-compact case (it is not even known how to detect the weight filtration). However in simple cases one can construct the plectic structure directly, using toroidal compactifications. In particular, one has the following result.

Theorem 16.2. Let G = R_{F/ℚ}(GL(2)_{F}) and 𝒳 = (ℂ ∖ ℝ)^{r}. Then the real cohomology of Y = Sh_{K}(G, 𝒳 ), for K ⊂ G(ℚ̂) a sufficiently small open compact subgroup, carries a canonical plectic mixed Hodge structure. The same is true for the mixed Shimura stack 𝒜 attached to G = R_{F/ℚ}(GL(2)_{F} ⋉ 𝔾_{a,F}^{2} ).

In fact, slightly more is true: by the methods of [4], there exists a complex \underline{RΓ}(Y, ℝ) equipped with filtrations W_{•} and (after tensoring with ℂ) F_{j}^{•}, whose cohomology is H^{∗}(Y, ℝ) together with its plectic Hodge structure. Then a similar procedure to [2] gives plectic absolute Hodge cohomology groups H_{ℋ ,plec}^{i} (Y, ℝ(rn)), together with a spectral sequence

E_{2}^{ij} = Ext_{ℝMHS_{plec}}^{i}(ℝ(−nr)_{plec}, H^{j}(Y, ℝ)) ⟹ H_{ℋ ,plec}^{i+j} (Y, ℝ(rn)).

Similar statements hold for the cohomology of 𝒜 and 𝒜 ∖ 𝒜 [d].

17. An arithmetic application

The formal procedure described in Section 13 may now be carried through in plectic absolute Hodge cohomology. One thus obtains canonical elements

dΘ_{ℋ ,plec} ∈ H_{ℋ ,plec}^{2r−1} (𝒜 ∖ 𝒜 [d], ℝ(r)),

together with their pull-backs by torsion sections arising from the level structure

x^{∗}(_{d}Θ_{ℋ ,plec} ) ∈ H_{ℋ ,plec}^{2r−1} ([Δ\Y], ℝ(r))

and, by taking the cap-product with a generator of H_{r−1}(Δ, ℤ), classes

g_{ℋ ,plec} = x^{∗}(_{d}Θ_{ℋ ,plec} ) ∩ η_{Δ} ∈ H_{ℋ ,plec}^{r} (Y, ℝ(r)),

which are the plectic analogues of the functions log |(Siegel unit)|.

There is an explicit formula for the form representing the class g_{ℋ ,plec} in terms of coordinates. For simplicity we give this in the case Y = Γ\ℌ^{r} for Γ ⊂ SL_{2}(O_{F}), parametrising abelian varieties A_{τ} = Λ_{τ} \ℂ^{r} for τ = (τ_{j}) ∈ ℌ^{r}, where ℌ is the upper half-plane and Λ_{τ} = 𝔡_{F}^{−1} +τO_{F}. The variety A_{τ} carries a canonical principal polarisation associated to the form ∑ dz_{j}∧dz̄_{j}/ Im(τ_{j}). Let (−, −) be the associated Hermitian form on ℂ^{r}. For γ ∈ Λ_{τ}, let χ_{γ}: A_{τ} → U(1) be the character given by Pontryagin duality and the polarisation form. The real analytic function g(τ) representing the class g_{ℋ ,plec} is then given by

g(τ) = (constant) ∑_{y∈A_{τ}[d]} \begin{matrix}∑ & ′ \\ γ \end{matrix}(χ_{γ}(x + y) − χ_{γ}(x)) ∏_{j=1}^{r} \frac{1}{γ_{j}γ̄_{j}} ,

the sum being taken over nonzero γ ∈ Λ_{τ} modulo Δ. When τ is a CM-point, taking linear combinations of finite Fourier transforms of these series one obtains special values L^{(r)}(χ, 0), where χ is an abelian character of the Galois group of the CM field F (τ).

Space precludes describing in detail here the Hodge-theoretic version of the cup product constructions sketched in Section 15, which lead to a formula for the r-th derivative L^{(r)}(f, 0), where f is a cuspidal Hilbert eigenform of parallel weight 2.

The plectic theta elements constructed above form the first of a hierarchy of cohomology classes, coming from a plectic analogue of the abelian polylogarithm classes of Wildeshaus [24]. There are explicit formulae for these higher plectic classes, related to those described by Levin [12]. Restricting to torsion sections, these polylogarithmic classes define plectic Eisenstein classes in the plectic absolute Hodge cohomology of Y (now with non-trivial coefficients), and evaluating these at CM points gives a cohomological interpretation of the derivatives L^{(r)}(χ, 0), for all algebraic Hecke characters χ of the CM field for which s = 0 is a non-critical value with ord_{s=0}L(χ, s) = r.

18. Final speculations

Suppose X is a Shimura variety or stack of the type considered above, or a diagram of such objects. In the presence of Conjecture 6.1, we have defined in 7.1 the plectic absolute étale cohomology groups H_{et,plec}^{∗}(X, ℚ_{ℓ}(n)) of X, for appropriate n ∈ ℤ. We also have (in many cases unconditionally) plectic absolute Hodge cohomology groups H_{ℋ ,plec}^{∗} (X, ℝ(n)). In view of the universal nature of (usual) motivic cohomology, one is led to speculate:

Question 18.1. Do there exist “plectic motivic cohomology groups” for X: H_{ℳ ,plec}^{∗} (X, ℤ(n)) or H_{ℳ ,plec}^{∗} (X, ℚ(n)), with good functorial properties, together with functorial maps to the plectic étale and absolute Hodge cohomology groups?

Suppose such groups did exist and were sufficiently functorial that the constructions of Sections 13–14 could be carried out. As indicated there, one could then prove versions of Stark’s conjectures for abelian characters of CM and totally real fields, as well as special cases of Beilinson’s conjectures for non-critical L-values of Hilbert modular forms.

The discussion in Sections 11–16 involved generalisations of “cyclotomic” zeta elements. It is tempting to ask:

Question 18.2. Is there a plectic analogue of anticyclotomic zeta-elements, such as Heegner points and (generalised) Heegner cycles?

Let f be a Hilbert modular eigenform over F and χ an algebraic Hecke character of a totally imaginary quadratic extension L of F (suitably compatible with the central character of f). It seems that a hypothetical geometric version (1.3) of the plectic formalism would yield interesting elements responsible for the special values of anticyclotomic Rankin-Selberg L-functions L(f × θ_{χ}, s) at the central point, in the case when the order of vanishing a of the L-function satisfies 1 ≤ a ≤ r. This will be discussed in more detail in a later paper in this series.

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Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie, 4 place Jussieu, F-75252 Paris, France

E-mail address: jan.nekovar@imj-prg.fr

Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, England

E-mail address: a.j.scholl@dpmms.cam.ac.uk