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An analogue of circular unit for products of elliptic curves. (English) Zbl 1069.11016

Let \(K/\mathbb Q\) be a real quadratic field with quadratic character \(\chi\). Suppose \(\chi\) has conductor \(N\). Then \(\chi\) determines a unit \(U_{\chi}\in \mathcal O^{\times}_{K}\), called the circular unit, by \[ U_{\chi} = \prod | 1-\xi^{k}| ^{-\chi (x)/2}, \] where the product is taken over residues \(k\) mod \(N\) with \((k,N)=1\), and where \(\xi = \exp (2\pi i/N)\). Then according to Dirichlet and Dedekind (?), we have \[ L' (0,\chi) = \log | U_{\chi}| , \tag{\(*\)} \] where \(L (s,\chi) = \sum_{n\geq 1} \chi (n)n^{-s}\) is the Dirichlet \(L\)-function attached to \(\chi\).
Now let \(X/\mathbb Q\) be a smooth projective variety of dimension \(\leq m\), and let \(L_{2m} (X,s)\) be the \(L\)-function attached to the (suitable) cohomology group \(H^{2m} (X)\). Then Beilinson has given a conjectural formula for \(L^{*}_{2m} (X,m)\), where the \(*\) means to take the first nonvanishing coefficient in the Taylor expansion of \(L_{2m}\) at \(s=m\). This formula generalizes \((*)\), in the sense that it reduces to \((*)\) when \(X=\text{Spec}\, K\) and \(m=0\).
The main results of the paper under review concern the case \(X=E\times E'\), where \(E,E'\) are two non-isogenous elliptic curves over \(\mathbb Q\). The motivating idea is that \(E\times E'\) is analogous to a real quadratic field; in this case the appropriate special value is \(L' (H^{1} (E)\otimes H^{1} (E'),1)\). The authors’ first result is that there is an element in the motivic cohomology \(H_{\mathcal M}^{3} (E\times E', \mathbb Q (2))\) that plays the role of the circular unit \(U_{\chi}\) in this analogy. The second result is an explicit formula for the special value in terms of a sum of certain integrals involving the elliptic modular forms associated to \(E\) and \(E'\). The paper concludes with a further discussion of the analogy between real quadratic fields and products of pairs of non-isogenous elliptic curves.

MSC:

11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
14G35 Modular and Shimura varieties
11F11 Holomorphic modular forms of integral weight
11E45 Analytic theory (Epstein zeta functions; relations with automorphic forms and functions)
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
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