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Cycles in a product of elliptic curves, and a group analogous to the class group. (English) Zbl 0788.14004
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.

##### MSC:
 14C05 Parametrization (Chow and Hilbert schemes) 14G35 Modular and Shimura varieties 14H52 Elliptic curves 14C25 Algebraic cycles
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