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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

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