## Abstract

Let f be a modular form of weight k≫2 and level N, let K be a quadratic imaginary field and assume that there is a prime p exactly dividing N. Under certain arithmetic conditions on the level N and the field K, one can attach to this data a p-adic L-function L p (f,K,s) , as done by Bertolini-Darmon-Iovita- Spie β in [Teitelbaum's exceptional zero conjecture in the anticyclotomic setting, Amer. J. Math. 124 (2002), 411-449]. In the case of p being inert in K, this analytic function of a p-adic variable s vanishes in the critical range s=1,...,k-1 , and one may be interested in the values of its derivative in this range. We construct, for k≫4 , a Chow motive endowed with a distinguished collection of algebraic cycles which encode these values, via the p-adic Abel-Jacobi map. Our main result generalizes the result obtained by Iovita and Spieβ in [Derivatives of p-adic L-functions, Heegner cycles and monodromy modules attached to modular forms, Invent. Math. 154 (2003), 333-384], which gives a similar formula for the central value s=k/2. Even in this case our construction is different from the one found by Iovita and Spieβ.

Original language | English |
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Pages (from-to) | 1003-1032 |

Number of pages | 30 |

Journal | Compositio Mathematica |

Volume | 148 |

Issue number | 4 |

DOIs | |

Publication status | Published - Jul 2012 |

## Keywords

- anti-cyclotomic p-adic L-function
- CM cycles
- p-adic integration
- Shimura curve