CANONICAL LIFTS OF FAMILIES OF ELLIPTIC CURVES
James Borger, Lance Gurney
Nagoya Mathematical Journal | CAMBRIDGE UNIV PRESS | Published : 2019
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 Erdoǧan.