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ON ${p}$-ADIC HEIGHTS IN FAMILIES OF ELLIPTIC CURVES

Published online by Cambridge University Press:  23 July 2004

CHRISTIAN WUTHRICH
Affiliation:
Trinity College, Cambridge, United Kingdomc.wuthrich@dpmms.cam.ac.uk
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Abstract

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 $\mathbb{Z}_p$-extensions.

Type
Notes and Papers
Copyright
The London Mathematical Society 2004

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