## Abstract

Let f be a primitive Hilbert modular form of parallel weight 2 and level N for the totally real field F, and let p be a rational prime coprime to 2N. If f is ordinary at p and E is a CM extension of F of relative discriminant Δ prime to Np, we give an explicit construction of the p-adic Rankin–Selberg L-function L_{p}(f_{E},.). When the sign of its functional equation is −1, we show, under the assumption that all primes ℘∣p are principal ideals of 6_{F} that split in 6_{E}, that its central derivative is given by the p-adic height of a Heegner point on the abelian variety A associated with f. This p-adic Gross–Zagier formula generalises the result obtained by Perrin-Riou when F=ℚ and (N,E) satisfies the so-called Heegner condition. We deduce applications to both the p-adic and the classical Birch and Swinnerton-Dyer conjectures for A.

Original language | English |
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Pages (from-to) | 1571-1646 |

Number of pages | 76 |

Journal | Algebra and Number Theory |

Volume | 9 |

Issue number | 7 |

DOIs | |

State | Published - 1 Jan 2015 |

Externally published | Yes |

## Keywords

- Gross–zagier
- Heegner points
- Hilbert modular forms
- P-adic L-functions
- P-adic heights

## ASJC Scopus subject areas

- Algebra and Number Theory