$p$-adic heights of Heegner points and Beilinson-Flach elements

Abstract

We give a new proof of Howard's $\Lambda$-adic Gross-Zagier formula, which we extend to the context of indefinite Shimura curves over $\mathbf{Q}$ attached to nonsplit quaternion algebras. This formula relates the cyclotomic derivative of a two-variable $p$-adic $L$-function restricted to the anticyclotomic line to the cyclotomic $p$-adic heights of Heegner points over the anticyclotomic tower, and our proof, rather than inspired by the original approaches of Gross-Zagier and Perrin-Riou, is via Iwasawa theory, based on the connection between Heegner points, Beilinson-Flach elements, and their explicit reciprocity laws.

To appear in J. Lond. Math. Soc

References in corpus (8)

• Rankin--Eisenstein classes and explicit reciprocity laws
• Rankin--Eisenstein classes for modular forms
• On the non-triviality of the $p$-adic Abel-Jacobi image of generalised Heegner cycles modulo $p$, II: Shimura curves
• Heegner Point Kolyvagin System and Iwasawa Main Conjecture
• The Birch and Swinnerton-Dyer Formula for Elliptic Curves of Analytic Rank One
• Rankin-Selberg Euler systems and p-adic interpolation
• Iwasawa Main Conjecture for Heegner Points: Supersingular Case
• Anticyclotomic p-ordinary Iwasawa Theory of Elliptic Modular Forms

Cited by in corpus (4)

• Comparing anticyclotomic Selmer groups of positive coranks for congruent modular forms -- Part II
• Statistics for Anticyclotomic Iwasawa Invariants of Elliptic Curves
• Anticyclotomic p-ordinary Iwasawa Theory of Elliptic Modular Forms
• Main conjectures for higher rank nearly ordinary families -- I