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Kolyvagin’s method for Chow groups and Kuga-Sato varieties. (English) Zbl 0729.14004
Let \(f\in S_{2r}^{new}(\Gamma_ 0(N))\) be a newform of weight 2r\(\geq 4\) with rational coefficients, M the \(\ell\)-adic representation of G(\({\bar {\mathbb{Q}}}/{\bar {\mathbb{Q}}})\) associated to f. It is a factor of \(H_{et}^{2r-1}(Y\otimes {\bar {\mathbb{Q}}},{\mathbb{Q}}_{\ell})\), where Y is a suitable smooth compactification of the (2r-2)-fold fibre product of the universal elliptic curve over the open modular curve Y(N). The \(\ell\)-adic Abel-Jacobi map (over any extension K of \({\mathbb{Q}})\) induces a map \[ \Phi:\;CH^ r(Y/K)_ 0\to H^ 1_{cont}(K,H_{et}^{2r- 1}(Y\otimes {\bar {\mathbb{Q}}},{\mathbb{Q}}_{\ell})(r))\to H^ 1_{cont}(K,M(r)), \] where \(CH^ r(Y/K)_ 0\) denotes the group of homologically trivial cycles on Y of codimension r defined over K, modulo rational equivalence. If K is an imaginary quadratic field in which all primes dividing N split, there is a Heegner cycle \(y\in CH^ r(Y/K)_ 0\otimes {\mathbb{Q}}.\)
Theorem. Suppose that \(\ell\) does not divide 2(2r-2)!N\(\phi\) (N). If \(\Phi\) (y) is nonzero, then \(Im(\Phi)\otimes {\mathbb{Q}}_{\ell}={\mathbb{Q}}_{\ell}.\Phi (y).\)
A similar statement is proved for newforms with not necessarily rational coefficients.
Reviewer: J.Nekovář

MSC:
14C05 Parametrization (Chow and Hilbert schemes)
11F11 Holomorphic modular forms of integral weight
14C15 (Equivariant) Chow groups and rings; motives
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