The p-adic Gross-Zagier formula on Shimura curves

Abstract

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.

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

References in corpus (1)

• Explicit Gross-Zagier and Waldspurger Formulae

Cited by in corpus (5)

• The universal $p$-adic Gross-Zagier formula
• The $p$-adic Gross-Zagier formula on Shimura curves, II: nonsplit primes
• Local Langlands correspondence, local factors, and zeta integrals in analytic families
• On the p-adic Birch and Swinnerton-Dyer conjecture for elliptic curves over number fields
• $p$-adic L-functions via local-global interpolation: the case of ${\mathrm{GL}}_2\times \mathrm{GU}(1)$