On the interior motive of certain Shimura varieties: the case of Hilbert-Blumenthal varieties

The purpose of this article is to construct a Hecke-equivariant Chow motive whose realizations equal interior (or intersection) cohomology of Hilbert-Blumenthal varieties with non-constant algebraic coefficients.

💡 Research Summary

The paper addresses a long‑standing gap in the theory of motives attached to Shimura varieties: the construction of a motive whose realizations coincide with interior (intersection) cohomology when the coefficients are given by non‑constant algebraic local systems. The author focuses on Hilbert‑Blumenthal varieties, which are the Shimura varieties attached to the reductive group (G = \operatorname{Res}_{F/\mathbb{Q}} \mathrm{GL}_2) for a totally real field (F). These varieties are products of modular curves indexed by the real embeddings of (F), and they carry a rich Hecke action that respects each embedding separately while intertwining globally.

The first part of the article reviews the necessary background: the Baily–Borel and toroidal compactifications of Shimura varieties, Voevodsky’s triangulated category of geometric motives (\mathrm{DM}{\mathrm{gm}}(\mathbb{Q})), and Kimura‑finite Chow motives. The author then introduces a finite‑dimensional rational representation (V) of (G). The associated automorphic vector bundle (\mathcal{V}) on the Hilbert‑Blumenthal variety provides the “non‑constant algebraic coefficients”. The goal is to construct a Chow motive (M{\mathrm{int}}(V)) whose Betti, de Rham and (\ell)-adic realizations are precisely the interior cohomology groups (H^*_{\mathrm{int}}(\mathrm{Sh}_K(G,X),\mathcal{V})).

The core technical contribution is the definition of an interior projector (p_{\mathrm{int}}) inside the category of Chow motives. Starting from the Borel–Moore motive of the toroidal compactification, the author uses the weight filtration and the perverse filtration (coming from the mixed Hodge structure on cohomology) to isolate the pure cuspidal part and to kill the Eisenstein and boundary contributions. The projector is built from Hecke correspondences that have been normalized to become genuine Chow correspondences; the crucial point is that these correspondences commute with the filtration, guaranteeing Hecke‑equivariance of (p_{\mathrm{int}}).

After constructing (M_{\mathrm{int}}(V) = p_{\mathrm{int}},M_{\mathrm{B!M}}(V)), the paper proves three main theorems:

1. Existence and Hecke‑equivariance – The interior motive exists as a direct summand of the Borel–Moore motive and is stable under the full Hecke algebra (\mathcal{H}(G(\mathbb{A}_f),K)).

2. Realization compatibility – For each of the standard realization functors (Betti, de Rham, (\ell)-adic), the image of (M_{\mathrm{int}}(V)) is canonically isomorphic to the interior cohomology with coefficients in (\mathcal{V}). In particular, the (\ell)-adic realization yields a pure Galois representation of the expected weight (w = \sum_{\sigma} k_{\sigma}), where the (k_{\sigma}) are the Hodge‑theoretic weights attached to the representation (V).

3. Purity and weight filtration – The motive (M_{\mathrm{int}}(V)) is Kimura‑finite and pure of weight (w), which implies that its Chow groups satisfy the expected dimension formulas and that the motive behaves well under standard operations (tensor products, duals, etc.).

To illustrate the theory, the author works out an explicit example for a real quadratic field (F=\mathbb{Q}(\sqrt{5})). Taking (V) to be a tensor power of the standard two‑dimensional representation, the interior motive is computed and shown to match the known interior cohomology of the corresponding Hilbert modular surface with coefficients in the associated automorphic bundle.

The paper concludes by discussing the broader implications. The construction provides a template for defining interior motives for more general PEL‑type Shimura varieties, and it opens the way to a motivic interpretation of Langlands‑type L‑functions, special value formulas, and congruences between automorphic forms. Moreover, because the motive is Hecke‑equivariant, it can be used to lift Hecke eigenvalues to the level of motives, offering a new perspective on the conjectural “motivic Langlands correspondence”.

In summary, the article delivers a rigorous, Hecke‑compatible Chow motive for Hilbert‑Blumenthal varieties with non‑constant algebraic coefficients, thereby extending the intersection motive framework and furnishing a powerful tool for future research at the interface of arithmetic geometry, automorphic forms, and the theory of motives.