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CANONICAL LIFTS OF FAMILIES OF ELLIPTIC CURVES

Part of: Arithmetic rings and other special rings Curves

Published online by Cambridge University Press:  02 November 2017

JAMES BORGER  and
LANCE GURNEY Open the ORCID record for LANCE GURNEY[Opens in a new window]
JAMES BORGER
Affiliation:
Mathematical Sciences Institute, Australian National University, Canberra ACT 0200, Australia email james.borger@anu.edu.au
LANCE GURNEY
Affiliation:
Korteweg-de Vries Instituut, Universiteit van Amsterdam, 1090 GE Amsterdam, The Netherlands email l.r.gurney@uva.nl
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Abstract

We show that the canonical lift construction for ordinary elliptic curves over perfect fields of characteristic $p>0$ extends uniquely to arbitrary families of ordinary elliptic curves, even over $p$-adic formal schemes. In particular, the universal ordinary elliptic curve has a canonical lift. The existence statement is largely a formal consequence of the universal property of Witt vectors applied to the moduli space of ordinary elliptic curves, at least with enough level structure. As an application, we show how this point of view allows for more formal proofs of recent results of Finotti and Erdoğan.

MSC classification

Primary: 13F35: Witt vectors and related rings 14H52: Elliptic curves
Type
Article
Information
Nagoya Mathematical Journal , Volume 233 , March 2019 , pp. 193 - 213
Copyright
© 2017 Foundation Nagoya Mathematical Journal  

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