Cycles in a product of elliptic curves, and a group analogous to the class group
type: Item
instance of: scholarly article
title: Cycles in a product of elliptic curves, and a group analogous to the class group
zbMATH Open document ID: 0788.14004
DOI: 10.1215/S0012-7094-92-06715-9
author: Stephen J. M. Mildenhall
published in: Duke Mathematical Journal
publication date: 1992-12-14T00:00:00Z
review text: 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 conjectur…
Mathematics Subject Classification ID: 14C05
Mathematics Subject Classification ID: 14G35
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 14C25
zbMATH DE Number: 78564
zbMATH Keywords: second Chow group of a surface
zbMATH Keywords: Picard number
zbMATH Keywords: modular elliptic curve
zbMATH Keywords: Bass conjecture
zbMATH Keywords: Beilinson conjectures
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 78564