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Higher regulators and Hecke \(L\)-series of imaginary quadratic fields. I. (English) Zbl 0721.14004
Bloch provided a \(K\)-theoretical interpretation of the value \(L(E,2)\) of the \(L\)-series associated with a CM elliptic curve \(E\) over \({\mathbb Q}\). His ideas have been extended by Beilinson to very general conjectures on special values of \(L\)-series of motives over number fields [A. A. Beilinson, J. Sov. Math. 30, 2036–2070 (1985); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 24, 181–238 (1984; Zbl 0588.14013)]. In this paper, we generalize Bloch’s result, in the context of the Beilinson conjectures, to all values of \(L(E,s)\) for \(s=2,3,..\). for certain elliptic curves \(E\), with complex multiplication, and defined over number fields. The precise family of elliptic curves which we consider is as follows. Let \(K\) be an imaginary quadratic field, and \(F\) a finite extension of \(K\). Denote by \(E\) an elliptic curve defined over \(F\) with \(End_ F(E)\) the maximal order in \(K\). We shall say that \(E\) satisfies condition (S) if:
The extension \(F(E_{tors})/K\) is abelian, where \(E_{tors}\) denotes the group of all torsion points on \(E\).
One of the main results of this paper is as follows: For \(\ell \geq 0\) consider the \(a^{\ell +2}\) eigenspace \(H^ 2_{{\mathcal M}}(E,{\mathbb Q}(\ell +2))\) for all Adams operators \(\psi^ a\) for \(a\geq 1\) on the algebraic \(K\)-group \(K_{2\ell +2}(E)\otimes {\mathbb Q}\). For these “motivic cohomology groups” we have Beilinson’s regular map: \[ r_{{\mathcal D}}:\;H^ 2_{{\mathcal M}}=H^ 2_{{\mathcal M}}(E,{\mathbb Q}(\ell +2))\to H^ 2_{{\mathcal D}}=\oplus_{\sigma \in G}H^ 1(^{\sigma}E({\mathbb C}),(2\pi i)^{\ell +1}{\mathbb R}), \] where \(\sigma\) runs over the \(K\)-embeddings of \(F\) into \({\mathbb C}\). If \(E\) satisfies condition (S) we will construct linearly independent (conjecturally a basis) \(\xi_ 1,...,\xi_{2g}\) in \(H^ 2_{{\mathcal M}}\) where \(g=(F:K)\) with the following property:
If \(\eta_ 1,...,\eta_{2g}\) is a basis of \(\oplus_{\sigma \in G}H^ 1(^{\sigma}E({\mathbb C}),(2\pi i)^{\ell +1}{\mathbb Q}) \) and if we write \(r_{{\mathcal D}}(\xi_ i)=\sum_{j}a_{ij}\eta_ j \) in \(H^ 2_{{\mathcal D}}\) for \(a_{ij}\in {\mathbb R}\) then \[ L(E,\ell +2)\equiv (2\pi)^{4g(\ell +1)}\det (a_{ij}) \mod {\mathbb Q}^*.\tag{0.2} \]
From the functional equation it follows that \(L(E,s)\) has a zero of order \(2g=\dim H^ 2_{{\mathcal D}}\) at \(s=-\ell\) and that \[ L^{(2g)}(E,- \ell)\equiv \det (a_{ij}) \mod{\mathbb{Q}}^*. \]
The case where \(E\) is defined over a real subfield of \(F\) will be considered in Part II of this series [Ann. Math. (2) 132, No. 1, 131–158 (1990; Zbl 0721.14005)].
If \(E\) has complex multiplication without satisfying condition (S) we can still calculate the regulator determinant \(\det (a_{ij})\) for certain elements \(\xi_ 1,...,\xi_{2g}\) in \(H^ 2_{{\mathcal M}}\). The \(a_{ij}\) are special values of \({\mathbb Q}\)-linear combinations of Eisenstein-Kronecker series. If \(\det (a_{ij})\) is non zero the Beilinson conjectures imply that (0.2) should hold in this case as well. In particular, the values \(L(E,s)\) for \(s=2,3,..\). should be related to nonlinear combinations of Eisenstein-Kronecker series.
If \(E\) does not have CM the method for \(\ell >0\) no longer provides elements in \(H^ 2_{{\mathcal M}}(E,{\mathbb{Q}}(\ell +2))\), but only in the motivic cohomology of symmetric powers of the motive of \(E\). The conjectural consequences for \(L(Sym^ n(H^ 1(E)),n+1)\) with \(n\geq 1\) will be considered in part II (loc. cit.).
It is believed that there exists a motivic cohomology theory with integral coefficients. Such a theory is necessary if one wants to form a regulator which is determined up to sign and not only up to a number in \({\mathbb Q}^*\). We have listed the required properties of motivic cohomology theory with integral coefficients in (1.4) and work in this context. We rely heavily on the general background given by C. Goldstein and N. Schappacher in J. Reine Angew. Math. 327, 184–218 (1981; Zbl 0456.12007). However, we deal differently with the \(L\)-series using the adèlic method.
In the known evidence for Beilinson’s conjectures (for higher \(K\)-groups) two principal difficulties have to be overcome: The elements which one constructs by taking cup products always lies in the motivic cohomology of some open variety (or simplicial variety) \(X\) whereas one needs them in some compactification \(\bar X.\)
The second general difficulty is to relate the class in the Deligne cohomology of \(\bar X\) to the \(L\)-series. In our situation the \(L\)-value is given by integrating a singular form (with worse than logarithmic singularities!) against a smooth form without compact support. We present a direct analytical-topological method to overcome this difficulty.

MSC:
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11R42 Zeta functions and \(L\)-functions of number fields
11R70 \(K\)-theory of global fields
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
19F27 Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects)
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