Higher regulators, Hilbert modular surfaces, and special values of \(L\)-functions
The paper consists of an introduction to set the stage and to announce the main result (and some of its shortcomings), six sections and references. The first section gives a brief overview of the construction of abelian schemes over Hilbert modular surfaces. In the second section the main object of study, the fundamental relative motive \(\mathcal V^{p,q}\), is introduced. Its various realizations are discussed. In particular, there is a comparison result between the Betti realization and intersection cohomology. The third section deals with the Betti and de Rham realizations. The fourth section gives an overview of \(L\)-functions. The results of Brylinski-Labesse and Chai are discussed. The exposition is quite detailed and technical. In the fifth section the main result (cf. \textit{infra}) is presented. The proof of this result takes quite some place. As usual an important role is played by Beilinson's Eisenstein map. This is explained in the sixth section. Let \(F/{\mathbb Q}\) be a real quadratic field with ring of integers \(\mathcal O_F\). Let \(G/{\mathbb Q} \hookrightarrow\text{Res}_{F/{\mathbb Q}}\text{GL}_{2,F}\) such that the maps \(G\rightarrow{\mathbb G}_m\), \({\mathbb G}_m\hookrightarrow\text{Res}_ {F/{\mathbb Q}}{\mathbb G}_{m,F}\) and \(\text{det}:\text{Res}_{F/{\mathbb Q}}\text{GL}_{2,F}\rightarrow \text{Res}_{F/{\mathbb Q}}{\mathbb G}_{m,F}\) lead to a Cartesian square. The group \(G\) gives rise to a Shimura variety \(S({\mathbb C})\) with a \(G({\mathbb A}_f)\) action. Let \(V^{p,q}=\text{Sym}^pV_2^{\vee}\otimes\text{Sym}^qV _2^{\vee}\), where \(V_2^{\vee}\) is the dual of the standard representation of \(\text{GL}_2\). It is known that intersection cohomology with respect to the Baily-Borel compactification decomposes into \(G({\mathbb A}_f)\)-isotypic components \(H^2_{\text{ét}}(\pi_f)\) which are \(\text{Gal} (\overline{\mathbb Q}/{\mathbb Q})\)-modules. An important result due to J. L.~Brylinski and J.-P.~Labesse says that, for almost all \(p\), the local \(L\)-functions \(L_p(s,H^2_{\text{ét}}(\pi_f))\) can be computed as special values of some automorphic \(L\)-function. This \(L\)-function has simple zeros at \(s=1-n\), \(p+2\leq n\leq p+q+2\) for certain \(\pi_f\), assuming \(p\geq q>0\). It turns out to be appropriate to work with relative motives. Let \(\mathcal A/S\) be the universal abelian scheme over \(S\). \(\mathcal A\) comes equipped with an \(\mathcal O_F\)-action. As a matter of fact, one is interested in constructing elements in \(H^{p+q+3}_{\mathcal M}(\mathcal A^{p+q},\overline{\mathbb Q}(n))\). A basic role is played by the direct summand \(\mathcal V^{p,q}\) of \(R(\mathcal A^{p+q}/S)\), built as suitable \(\text{Sym}^p\otimes\text{Sym}^q\). In Beilinson's philosophy, for a pure (Chow) motive \(M\) with coefficients in \(\overline{\mathbb Q}\) and \(L\)-function \(L(s,H^i_{ \text{ét}}(M))\), one has, for \(n>i/2+1\), regulator maps \[ r_{\mathcal H}:H^{i+1} _{\mathcal M}(M_{\mathbb Z},{\mathbb Q}(n))\rightarrow H^{i+1}_{\mathcal H}(M\otimes {\mathbb R},{\mathbb R}(n)) \] such that there exist \(\overline{\mathbb Q}\)-subspaces \(\mathcal K(i,n) \subset H^{i+1}_{\mathcal M}(M_{\mathbb Z},{\mathbb Q}(n))\) such that \[ r_{\mathcal H} (\mathcal K(i,n))=L(n,H^i_{\text{ét}}(M))\det\mathcal{DR}(i,n), \] where \(\mathcal{DR} (i,n)\) is Deligne's \(\overline{\mathbb Q}\)-structure. In the underlying situation, write \(r_{\mathcal H}:H^{\bullet}_{!\mathcal M}(\mathcal V^{p,q},{\mathbb Q}(*))\rightarrow H^{\bullet}_{!\mathcal H}(\mathcal V^{p,q}\otimes{\mathbb R},{\mathbb R}(*))\) for Beilinson's regulator map. For \(S=S(\pi)\cup\{\ell\}\), let \({L_S(s,H^2_{\text{ét}} (\pi_f)^{\vee}))=\prod_{p\not\in S}L_p(s,H^2_{\text{ét}}(\pi_f)^{\vee})}\). The main result may be stated as follows: Let \(\pi\in\text{coh} (V^{p,q})\) be stable or \(\varepsilon(\pi_f)=-1\); then there is a \(G({\mathbb A}_f)\) -submodule \(\mathcal K(p,q,n)\subset H^{p+q+3}_{\mathcal M}(\mathcal V^{p,q}, {\mathbb Q}(n))\) for all \(p+2\leq n\leq p+q+2\) and \(p\geq q>0\), such that: (i) \(r_{\mathcal H}(\mathcal K(p,q,n))\subset H^{p+q+3}_{\mathcal H}(\mathcal V^{p,q}\otimes {\mathbb R},{\mathbb R}(n))\); (ii) \(r_{\mathcal H}(\mathcal K(p,q,n))(\pi_f)=L_S(n,H^2_ {\text{ét}}(\pi_f))\cdot\mathcal{DR}(p,q,n)(\pi_f)\) (equality of \(\overline{\mathbb Q}\)-subvector spaces in \({\mathbb R}\otimes_{\mathbb Q}\overline{\mathbb Q}\)). \(K(p,q,n)\) can be explicitly constructed. In (ii) \(\mathcal{DR}(p,q,n)(\pi_f)\) is Deligne's \(\overline{\mathbb Q}\)-structure on absolute Hodge cohomology \(H^{p+q+3}_{!\mathcal H}(\mathcal V^{p,q}\otimes{\mathbb R}, {\mathbb R}(n))(\pi_f)\).
scientific article; zbMATH DE number 1425091
scientific article