• DOI:10.1007/s00209-020-02522-8
• Corpus ID: 173991030

Generalized Heegner cycles on Mumford curves

@article{Longo2019GeneralizedHC,
  title={Generalized Heegner cycles on Mumford curves},
  author={Matteo Longo and Maria Rosaria Pati},
  journal={Mathematische Zeitschrift},
  year={2019},
  volume={297},
  pages={483-515}
}

• M. Longo, Maria Rosaria Pati
• Published 1 June 2019
• Mathematics
• Mathematische Zeitschrift

We study generalised Heegner cycles, originally introduced by Bertolini–Darmon–Prasanna for modular curves in Bertolini et al. (Duke Math J 162(6):1033–1148, 2013), in the context of Mumford curves. The main result of this paper relates generalized Heegner cycles with the two variable anticyclotomic p -adic L -function attached to a Coleman family $$f_\infty $$ f ∞ and an imaginary quadratic field K , constructed in Bertolini and Darmon (Invent Math 168(2):371–431, 2007) and Seveso (J Reine Angew Math 686:111–148, 2014). While in Bertolini and Darmon (Invent Math 168(2):371–431, 2007) and Seveso (J Reine Angew Math 686:111–148, 2014) only the restriction to the central critical line of this 2 variable p -adic L -function is considered, our generalised Heegner cycles allow us to study the restriction of this function to non-central critical lines. The main result expresses the derivative along the weight variable of this anticyclotomic p -adic L -function restricted to non necessarily central critical lines as a combination of the image of generalized Heegner cycles under a p -adic Abel–Jacobi map. In studying generalised Heegner cycles in the context of Mumford curves, we also obtain an extension of a result of Masdeu (Compos Math 148(4):1003–1032, 2012) for the (one variable) anticyclotomic p -adic L -function of a modular form f and K at non-central critical integers.