## Abstract

We prove a general formula for the -adic heights of Heegner points on modular abelian varieties with potentially ordinary (good or semistable) reduction at the primes above . The formula is in terms of the cyclotomic derivative of a Rankin-Selberg -adic -function, which we construct. It generalises previous work of Perrin-Riou, Howard, and the author to the context of the work of Yuan-Zhang-Zhang on the archimedean Gross-Zagier formula and of Waldspurger on toric periods. We further construct analytic functions interpolating Heegner points in the anticyclotomic variables, and obtain a version of our formula for them. It is complemented, when the relevant root number is rather than , by an anticyclotomic version of the Waldspurger formula. When combined with work of Fouquet, the anticyclotomic Gross-Zagier formula implies one divisibility in a -adic Birch and Swinnerton-Dyer conjecture in anticyclotomic families. Other applications described in the text will appear separately.

Original language | English |
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Pages (from-to) | 1987-2074 |

Number of pages | 88 |

Journal | Compositio Mathematica |

Volume | 153 |

Issue number | 10 |

DOIs | |

State | Published - 1 Oct 2017 |

Externally published | Yes |

## Keywords

- Birch and Swinnerton-Dyer conjecture
- Gross-Zagier formula
- Heegner points
- p-adic L-function
- p-adic heights

## ASJC Scopus subject areas

- Algebra and Number Theory