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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 Étale cohomology, higher regulators, zeta and $$L$$-functions ($$K$$-theoretic aspects)
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