2015/10/31 by Daniel Disegni · 3 citations
Mathematics · #Abelian group #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Conjecture #Context (archaeology) #Divisibility rule #Modular form #Shimura variety #math.NT #msc:11G40
paper · pdf · doi:10.1112/s0010437x17007308
published as Compositio Math. 153 (2017) 1987-2074 · 75 pages. The present version is identical to the previous one (and to the published version), except for footnotes signalling that the main theorem is off by a factor of 2. A list of errata is contained in the author's "The p-adic Gross-Zagier formula on Shimura curves, II", Appendix B
openalex created_date 2016/06/24 · openalex publication_date 2017/07/11 · arxiv created 2019/07/30 · arxiv updated 2019/07/31 · openalex updated_date 2026/08/05
We prove a general formula for the p -adic heights of Heegner points on modular abelian varieties with potentially ordinary (good or semistable) reduction at the primes above p . The formula is in terms of the cyclotomic derivative of a Rankin–Selberg p -adic L -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 +1 rather than -1 , by an anticyclotomic version of the Waldspurger formula. When combined with work of Fouquet, the anticyclotomic Gross–Zagier formula implies one divisibility in a p -adic Birch and Swinnerton-Dyer conjecture in anticyclotomic families. Other applications described in the text will appear separately.