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Higher regulators of modular curves. (English) Zbl 0609.14006
Applications of algebraic K-theory to algebraic geometry and number theory, Proc. AMS-IMS-SIAM Joint Summer Res. Conf., Boulder/Colo. 1983, Part I, Contemp. Math. 55, 1-34 (1986).
[For the entire collection see Zbl 0588.00014.]
Suppose $$M_{(N)}$$ is the moduli space for elliptic curves with level-N- structure, $$\bar M{}_{(N)}$$ its canonical compactification and $$\pi_{(N)}: X_{(N)}\to M_{(N)}$$ the universal curve. Further, consider $$M=\lim_{\leftarrow}(M_{(N)})$$, $$X=\lim_{\leftarrow}(X_{(N)})$$ as well as $$X^ i$$, with $$X^ i=X\times_ M\times...\times_ MX$$ the i-fold fibre product. If $$H^{\bullet}_{{\mathcal A}}$$ denotes the absolute cohomology, then one has a map $\pi_*^ i\circ \{, \}: H_{{\mathcal A}}^{i+1}(X^ i,{\mathbb{Q}}(i+1))\times H_{{\mathcal A}}^{i+1}(X^ i,{\mathbb{Q}}(i+1))\to H^ 2_{{\mathcal A}}(M,{\mathbb{Q}}(i+2)),$ with $$\{$$, $$\}$$ the Steinberg symbol and $$\pi_*^ i$$ the Gysin map.
Define $$H^ 2_{{\mathcal A}}(M,{\mathbb{Q}}(i+2))^{parab}:=im(\pi_*^ i\circ \{, \})$$ and $$H^ 2_{{\mathcal A}}(\bar M,{\mathbb{Q}})(i+2))^{parab}:=H^ 2_{{\mathcal A}}(M,{\mathbb{Q}}(i+2))^{parab}\cap H^ 2_{{\mathcal A}}(\bar M,{\mathbb{Q}}(i+2))$$. (For $$i\geq 0$$ the restriction map $$H^ 2_{{\mathcal A}}(\bar M,{\mathbb{Q}}(i+2))\to H^ 2_{{\mathcal A}}(M,{\mathbb{Q}}(i+2))$$ is an injection, for $$i>0$$ this induces an isomorphism of the parabolic parts.)
If $$i\geq 0$$, then $$H^ 2_{{\mathcal A}}(\bar M,{\mathbb{Q}}(i+2))^{parab}\subset H^ 2_{{\mathcal A}}(\bar M,{\mathbb{Q}}(i+2))_{{\mathbb{Z}}}$$, where $$H^ 2_{{\mathcal A}}(\bar M,{\mathbb{Q}}(i+2))_{{\mathbb{Z}}}$$ comes from a regular model $$\bar M/{\mathbb{Z}}$$. As main theorem the author proves that the image $$P=r(H^ 2(\bar M,{\mathbb{Q}}(i+2))^{parab}\otimes {\mathbb{Q}})$$ of the regulator map in $$H^ 1_ B(\bar M,{\mathbb{R}}(i+1))\otimes_{{\mathbb{Q}}}{\bar {\mathbb{Q}}}$$ has a $$GL_ 2({\mathbb{A}}_ f)$$ decomposition $$P=\oplus_{v}V\otimes P_ v,\quad such$$ that $$P_{V_ d}=\ell (V,-i)\cdot H^ 1_ B(M_ V,{\mathbb{Q}}(i+1))$$, with $$\ell (V,-i):=(d/ds)L(V,s)|_{s=-i}$$.
Reviewer: M.Heep

##### MSC:
 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry 14H25 Arithmetic ground fields for curves 18F25 Algebraic $$K$$-theory and $$L$$-theory (category-theoretic aspects) 11F27 Theta series; Weil representation; theta correspondences