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Journal of the London Mathematical Society 2004 70(1):23-40; doi:10.1112/S0024610704005277
© 2004 by London Mathematical Society
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© The London Mathematical Society

On p-Adic Heights in Families of Elliptic Curves

Christian Wuthrich

Trinity College Cambridge, United Kingdom, c.wuthrich{at}dpmms.cam.ac.uk

Received 11 July 2003. Revision received 2 December 2003.

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 Zp-extensions.


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