Abstract
Let \({\mathbf{{f}}}\) be a \(p\)-ordinary Hida family of tame level \(N\), and let \(K\) be an imaginary quadratic field satisfying the Heegner hypothesis relative to \(N\). By taking a compatible sequence of twisted Kummer images of CM points over the tower of modular curves of level \(\Gamma _0(N)\cap \Gamma _1(p^s)\), Howard has constructed a canonical class \(\mathfrak{Z }\) in the cohomology of a self-dual twist of the big Galois representation associated to \({\mathbf{{f}}}\). If a \(p\)-ordinary eigenform \(f\) on \(\Gamma _0(N)\) of weight \(k>2\) is the specialization of \({\mathbf{{f}}}\) at \(\nu \), one thus obtains from \(\mathfrak{Z }_{\nu }\) a higher weight generalization of the Kummer images of Heegner points. In this paper we relate the classes \(\mathfrak{Z }_{\nu }\) to the étale Abel-Jacobi images of Heegner cycles when \(p\) splits in \(K\).
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Notes
As defined in [29, (1.3.7)].
Notice the effect of the Tate twist on the filtrations.
Notice that our indices differ from those in [1].
So that \(\mathbb{T }^*\otimes _{\mathbb{I }}F_\nu \cong V_{{\mathbf{{f}}}_\nu }\) for every \(\nu \in \mathcal{X }_\mathrm{arith}(\mathbb{I })\).
As opposed to \(\omega ^{\frac{p-1}{2}}.\)
That \(\Omega _{\nu }^{(\eta )}\), which a priori just lies in \(F_\nu \), is indeed a unit is shown in [31, Prop. 6.4].
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Acknowledgments
It is a pleasure to thank my advisor, Prof. Henri Darmon, for suggesting that I work on this problem, and for sharing with me some of his wonderful mathematical insights. I thank both him and Adrian Iovita for critically listening to me while the results in this paper were being developed, and also Jan Nekovář and Victor Rotger for encouragement and helpful correspondence. It is a pleasure to acknowledge the debt that this work owes to Ben Howard, especially for pointing out an error in an early version of this paper, and for providing several helpful comments and corrections. Finally, I am very thankful to an anonymous referee whose valuable comments and suggestions had a considerable impact on the final form of this paper.
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Castella, F. Heegner cycles and higher weight specializations of big Heegner points. Math. Ann. 356, 1247–1282 (2013). https://doi.org/10.1007/s00208-012-0871-4
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DOI: https://doi.org/10.1007/s00208-012-0871-4