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Torsion zero-cycles on the self-product of a modular elliptic curve. (English) Zbl 0880.14001
Let $$E$$ be a modular elliptic curve defined over $$\mathbb{Q}$$ with the conductor $$N$$, and let $$X= E\times_\mathbb{Q} E$$, which is by definition a projective smooth surface over $$\mathbb{Q}$$. Let $$\text{CH}_0 (X)$$ be the Chow group of zero-cycles on $$X$$ modulo rational equivalence. Fix a prime $$p$$ and let $\rho_p: \text{CH}_0 (X)\{p\} \to H^4_{\text{cont}} (X,\mathbb{Z}_p (2))$ be the cycle map, where $$\text{CH}_0 (X)\{p\} \subset \text{CH}_0 (X)$$ is the subgroup of the $$p$$-primary torsion elements and the group on the right-hand side is the continuous $$p$$-adic étale cohomology group. The main result of the paper is the following.
Theorem 1. Assume $$E$$ has no complex multiplication over any finite extension of $$\mathbb{Q}$$ and that $$N$$ is square-free and $$p\nmid 6N$$. Then $$\rho_p$$ is injective.
We remark that the assumption $$p\nmid 6$$ is due to a certain technical problem in the $$p$$-adic Hodge theory. The assumption $$p\nmid N$$ is more essential. As a corollary one obtains the following.
Theorem 2. Let the assumption be as in theorem 1. Then $$\text{CH}_0 (X) \{p\}$$ is finite.
We have the following additional result.
Theorem 3. There exists a finite set $$S$$ of rational primes for which we have $$\text{CH}_0 (X)\{p\} =0$$ if $$p\notin S$$.
There have been quite a few results on the finiteness of the torsion in the Chow group of algebraic cycles in codimension two on a projective smooth variety $$Y$$ over a number field. The above result in theorem 2 is the first case of the finiteness without the assumption $$H^2 (Y,{\mathcal O}_Y) =0$$.
The key to the proof of theorem 1 are the works of S. J. M. Mildenhall [Duke Math. J. 67, No. 2, 387-406 (1992; Zbl 0781.14004)] and M. Flach [Invent. Math. 109, No. 2, 307-327 (1992; Zbl 0788.14022)], where they constructed certain elements in $$H^1_{Zar} (X, {\mathcal K}_2)$$, a $$K$$-cohomology group of $$X$$, by using the theory of modular curves and modular forms. Using those elements Mildenhall was able to show that for each fixed integer $$\nu>0$$, $$_{p^\nu} \text{CH}_0(X): =\{c\in \text{CH}_0 (X) \mid p^\nu \cdot c= 0\}$$ is finite. On the other hand, the main result of M. Flach (loc. cit.) concerns the finiteness of the Selmer group $S(\mathbb{Q},A) =\text{Ker} \left(H^1 (\mathbb{Q},A) \to\bigoplus_{\text{all }\ell} {H^1 (\mathbb{Q}_\ell,A) \over H^1_f (\mathbb{Q}_\ell,A)} \right)$ associated to the $$\text{Gal} (\overline\mathbb{Q}/ \mathbb{Q})$$-module $$A=H^2 (\overline X, \mathbb{Q}_p/ \mathbb{Z}_p (2))$$, where $$\ell$$ ranges over all prime numbers and $$\mathbb{Q}_\ell$$ is the field of $$\ell$$-adic numbers. Finally, we note the following consequence of theorem 1, which gives modest evidence of a general conjecture of S. Bloch [Ann. Math., II. Ser. 114, 229-265 (1981; Zbl 0512.14009); conjecture (3-16)].
Theorem 4. Let $$\coprod \text{CH}_0 (X)= \text{Ker} (\text{CH}_0 (X)\to \prod_{\text{all} \ell} \text{CH}_0 (X\times_\mathbb{Q} \mathbb{Q}_\ell))$$. Let the assumption be as in theorem 1. Then the $$p$$-primary torsion part of $$\coprod \text{CH}_0 (X)$$ is a subquotient of $$S (\mathbb{Q},A)$$.

##### MSC:
 14C05 Parametrization (Chow and Hilbert schemes) 14H52 Elliptic curves 14C15 (Equivariant) Chow groups and rings; motives 14H25 Arithmetic ground fields for curves 14G35 Modular and Shimura varieties 11G05 Elliptic curves over global fields
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