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Extensions of motives of modular forms. (English) Zbl 0913.11025
Write \(\mathcal{E}\) for the universal elliptic curve, and let \(p_m^n:\mathcal{E}^n\rightarrow\mathcal{E}^m\) (resp. \(q_m^n:\mathcal{E}^n\rightarrow\mathcal{E}^m\)) be the projection onto the first (resp. last) \(m\) fibre coordinates if \(n\geq m\) and inclusion of the first (resp. last) \(n\) fibre coordinates times \((m-n)\)-times the identity section if \(n<m\). With the notations of G. Kings [J. Reine Angew. Math. 503, 109-128 (1998; see the preceding review)], for elements \(\varphi\in\text{Hom}_{\mathcal{MM}}(\overline{\mathbb Q}(0),\underline{H}^{r+1}(\mathcal{E}^r,r+1)(\varepsilon_r))\) and \(\psi\in\text{Hom}_{\mathcal{MM}}(\overline{\mathbb Q}(0),\underline{H}^{s+1}(\mathcal{E}^s,s+1)(\varepsilon_s))\), one constructs an extension \[ \mathcal{F}_{r,s}(\varphi\otimes\psi)\in\text{Ext}^1_{\mathcal{MM}}(\overline{\mathbb Q}(0),\underline{H}^{k+1}(\mathcal{E}^k,k+l+2)(\varepsilon_k)), \] where \(r+s=k+2l\). Writing \(\tau\) for the horospherical isomorphism, one has a commutative diagram \[ \begin{matrix} \Phi({\mathbb A}_f^2)_{-k}\otimes\nu^{\otimes k}& {\buildrel\text{Eis}_{\mathcal{M}}\over\longrightarrow}&H_{\mathcal{M}} ^{k+1}(\mathcal{E}^k,\overline{\mathbb Q}(k+1))(\varepsilon_k)\\ \tau\downarrow \simeq&&\downarrow\text{ch}\\ \bigoplus_{\eta=\eta_{\infty}\otimes\eta_f}\text{Ind}_{B({\mathbb A}_f)}^{\text{GL}_2({\mathbb A}_f)}\overline{\mathbb Q}(\eta_f)&{\buildrel \text{Eis}\over\longrightarrow}&\text{Hom}_{\mathcal{MM}}(\overline{\mathbb Q}(0),\underline{H}^{k+1}(\mathcal{E}^k,k+1)(\varepsilon_k)). \end{matrix} \] The main result now says: Let \(r+s=k+2l\) with \(r\geq s\geq l\) and let \(f\in\Phi({\mathbb A}_f^2)_{-r}\otimes\nu^{\otimes r}\), \(g\in\Phi({\mathbb A}_f^2)_{-s}\otimes\nu^{\otimes s}\), then if \(s\neq 0\), \[ \mathcal{F}_{r,s}(\text{Eis }\tau(f)\otimes\text{Eis }\tau(g))=\text{ch}((p^{k+l}_{r-l}\times q^{k+l}_{s-l})_*(p^{k+l*}_r\text{Eis}_{\mathcal{M}}f\cup q^{k+l*}_s\text{Eis}_{\mathcal{M}} g)). \] If \(s=0\) and \(\psi\in H^1_{\mathcal{M}}(M_s,\overline{\mathbb Q}(1))\) one has \[ \mathcal{F}_{k,s}(\text{Eis}\tau(f)\otimes\text{ch}(\psi))=\text{ch}(\text{Eis}_{\mathcal{M}}f\cup q_0^{k*}\psi). \] Here \(M=\lim M_N\) and \(M_s:=M\cup s(\text{Spec}({\mathbb Q}(\mu_{\infty}))\) for a section \(s\) of the natural map \(M^{\infty}\rightarrow\text{Spec}({\mathbb Q}(\mu_{\infty}))\). As a corollary one obtains Beilinson’s conjecture for \(\underline{H}^{k+1}(\pi_f)\) where \(\pi=\pi_{\infty}\otimes\pi_f\) is a holomorphic cuspidal representation of weight \(k+2\).

MSC:
11G18 Arithmetic aspects of modular and Shimura varieties
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
19F27 √Čtale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects)
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