Variation of Heegner points in Hida families
By Benjamin Howard. 2006
88 citations recorded by the source corpus.
Given a weight two modular form f with associated p-adic Galois representation Vf, for certain quadratic imaginary fields K one can construct canonical classes in the Galois cohomology of Vf by taking the Kummer images of Heegner points on the modular abelian variety attached to f. We show that these classes can be interpolated as f varies in a Hida family and construct an Euler system of big Heegner points for Hida’s universal ordinary deformation of Vf. We show that the specialization of this big Euler system to any form in the Hida family is nontrivial, extending results of Cornut and Vatsal from modular forms of weight two and trivial character to all ordinary modular forms, and propose a horizontal nonvanishing conjecture for these cohomology classes. The horizontal nonvanishing conjecture implies, via the theory of Euler systems, a conjecture of Greenberg on the generic ranks of Selmer groups in Hida families.
Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/598ee47ad2a02fdd03fae180ef3705ffefe8216a
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Citations in the corpus
- Heegner cycles and p-adic L-functions (2015, methodology): “fTheoremA was provenin [Cas14]. The strategy followed in loc.cit. is the same as in this paper, but with our classes zf replaced by the specializations νf(Z∞) of Howard’s system of big Heegner points [How07] attached to the Hida family passing through f. In particular, a key ingredient in [Cas14] is”