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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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