On an elliptic analogue of Zagier's conjecture
Let \({\mathcal E}\to B\) be a smooth family of elliptic curves over a smooth quasiprojective connected scheme \(B\) over a number field \(K\) and \(i:B\to{\mathcal E}\) be its zero section. It is well-known that the group of sections \({\mathcal E}(B)\) is isomorphic to \(\text{Pic}_{{\mathcal E}/B}^0\) and the latter is isomorphic to the \((-1)\)-eigensubspace \(H_{\mathcal M}^2({\mathcal E},\mathbb{Q}(1))_-\) of the second motivic cohomology \(H_{\mathcal M}^2({\mathcal E},\mathbb{Q}(1))\) with respect to the action of the inversion automorphism. The author states a conjecture generalizing this fact which concerns higher motivic cohomology \(H_{\mathcal M}^{k-1}({\mathcal E}^{(k-2)},\mathbb{Q}(k-1))\cong K_{k-1}({\mathcal E}^{(k-1)}\otimes\mathbb{Q})^{k-1}\), \(k\geq 2\), where \({\mathcal E}^{(k-2)}= \text{Ker} (\sum:{\mathcal E}^{k-1}\to {\mathcal E})\). He also proves some partial results towards the conjecture. Let \({\mathcal L}_k\) be a vector space over \(\mathbb{Q}\) with basis \(\{s\}_k\), \(s\in {\mathcal E}(B)\setminus i(B)\) and let \[ D_k:{\mathcal L}_k\to{\mathcal L}_{k-1}\otimes_\mathbb{Q}({\mathcal E}(B)\otimes\mathbb{Q}) \] be a linear map defined by \(d_k(\{s\}_k)=\{s\}_{k-1}\otimes (s\otimes 1)\). The conjecture claims the existence of a homomorphism \(\text{Ker}(d_k)\to H_{\mathcal M}^{k-1}({\mathcal E}^{(k-2)},\mathbb{Q}(k-1))_{\text{sgn}}\) satisfying some functorial and naturality properties which we omit. Here the subscript sign denotes the eigensubspace of the sign character with respect to the natural action of the permutation group \(\Sigma_{k-1}\) on \({\mathcal E}^{(k-1)}\). When \(B=\text{Spec}(K)\), the author proves the torsion part of this conjecture by constructing the homomorphism defined on the subspace generated by the symbols of torsion sections. He also proves the conjecture for \(k=2\).
scientific article; zbMATH DE number 1026544
scientific article