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p ‐adic heights of Heegner points and Beilinson–Flach classes
Author(s) -
Castella Francesc
Publication year - 2017
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms.12058
Subject(s) - mathematics , iwasawa theory , context (archaeology) , pure mathematics , tower , quaternion , reciprocity law , reciprocity (cultural anthropology) , algebra over a field , geometry , psychology , paleontology , social psychology , abelian group , civil engineering , engineering , biology
We give a new proof of Howard's Λ ‐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 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 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.