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On p ‐Adic Heights in Families of Elliptic Curves
Author(s) -
Wuthrich Christian
Publication year - 2004
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/s0024610704005277
Subject(s) - pairing , mathematics , elliptic curve , degeneracy (biology) , valuation (finance) , algebraic number field , pure mathematics , physics , quantum mechanics , bioinformatics , superconductivity , finance , economics , biology
The non‐degeneracy of the canonical p‐adic height pairing defined by Perrin‐Riou and Schneider on an elliptic curve over a number field with good, ordinary reduction is still unknown. Following the work done for the real‐valued pairing, the behaviour of the p‐adic height is analysed as a point varies on a section of a family of elliptic curves, and so new information is obtained about this pairing. In particular, the variation is p‐adically continuous and the non‐degeneracy of a set of sections can be checked simultaneously for almost all elements of the family. The paper contains some conjectures about the valuation of the p‐adic regulator and its consequences for the growth of the Mordell–Weil group in cyclotomic Z p ‐extensions.