Cycles in a product of elliptic curves, and a group analogous to the class group
We consider a subgroup of the second Chow group of a surface, which we call \(\Sigma\). The group \(\Sigma\) is interesting because it is analogous to the divisor class group of classical number theory. It is defined as follows. Let \(X\) be a smooth algebraic surface defined over a number field \(K\) and suppose that \(X\) has a smooth proper model \({\mathcal X}\) defined over \({\mathfrak O}[1/N]\), a localization of the ring of integers in \(K\). Let \(j:X\to{\mathcal X}\) be the inclusion of \(X\) into \({\mathcal X}\). Then \(\Sigma\) is the kernel of the flat pull-back map \(j^*\): \(\Sigma:=\ker(j^*:\text{CH}^ 2({\mathcal X})\to\text{CH}^ 2(X))\). -- In this paper we give examples of surfaces with finite \(\Sigma\), but where the Picard number of the special fibre is strictly greater than the Picard number of the generic fibre for infinitely many primes. To do this requires elements in \(H^ 1(X,{\mathcal K}_ 2)\) which do not come from \(K^ \times\bigotimes_ \mathbb{Z}\text{Pic}(X)\). We show \(\Sigma\) is torsion in other similar examples. -- Our main theorem is the following one. Let \(E\) be an elliptic curve defined over \(\mathbb{Q}\) with complex multiplication by the whole ring of integers in the quadratic imaginary number field \(K\). Let \(N\) be the discriminant of \(E\), and \({\mathcal E}\) be the smooth curve over \({\mathfrak O}[1/6N]\) defined by \(E\). Then the kernels \[ \Sigma_ K=\ker(\text{CH}^ 2({\mathcal E}\times{\mathcal E})\to\text{CH}^ 2(E_ K\times E_ K)) \] and \[ \Sigma=\ker(\text{CH}^ 2({\mathcal E}_{\mathbb{Z}[1/6N]}\times{\mathcal E}_{\mathbb{Z}[1/6N]})\to\text{CH}^ 2(E\times E)) \] are finite. If \(E\) is defined over \(\mathbb{Q}\) and has a modular parametrization but no complex multiplication, then we can prove the following theorem: Let \(E\) be a modular elliptic curve defined over \(\mathbb{Q}\). Then \(\Sigma=\ker(j^*:\text{CH}^ 2{\mathcal X}_{\mathbb{Z}[1/5N]})\to\text{CH}^ 2(X))\) is torsion. There are several conjectural reasons to expect \(\Sigma\) to be torsion or finite, which we now briefly discuss. Bass conjecture, that \(K_ 0({\mathcal Y})\) is finitely generated for \({\mathcal Y}\) a regular scheme of finite type over \(\mathbb{Z}\), together with the Grothendieck isomorphism \(K_ 0({\mathcal Y})\times\mathbb{Q}\simeq(\bigoplus_ i\text{CH}^ i({\mathcal Y}))\otimes\mathbb{Q}\), would imply that \(\Sigma\) has finite rank. If \(\text{CH}^ 2({\mathcal Y})\) is finitely generated, then \(\Sigma\) would be finitely generated. -- An \(S\)-integral version of Beilinson's conjectures would imply that \(\Sigma\) is torsion. -- Finally, we mention that Bloch- Kato's Tamagawa number conjectures imply that \(\Sigma\) is finite.
scientific article; zbMATH DE number 78564
scientific article