p‐adic heights of Heegner points and Beilinson–Flach classes

Abstract

Abstract We give a new proof of Howard's ‐adic Gross–Zagier formula Compos. Math . 141 (2005) 811–846. MR 2148200 (2006f:11074)], which we extend to the context of indefinite Shimura curves over attached to nonsplit quaternion algebras. This formula relates the cyclotomic derivative of a two‐variable ‐adic ‐function restricted to the anticyclotomic line to the cyclotomic ‐adic heights of Heegner points over the anticyclotomic tower, and our proof, rather than inspired by the influential approaches of Gross–Zagier [ Invent. Math . 84 (1986) 225–320. MR 833192 (87j:11057)] and Perrin‐Riou [ Invent. Math . 89 (1987) 455–510. MR 903381 (89d:11034)], is via Iwasawa theory, based on the connection between Heegner points, Beilinson–Flach elements, and their explicit reciprocity laws.