The p-adic Gross-Zagier formula on Shimura curves

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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 languageAmerican English
Pages (from-to)1987-2074
Number of pages88
JournalCompositio Mathematica
Issue number10
StatePublished - 1 Oct 2017
Externally publishedYes


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

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory


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