Canonical lifts of families of elliptic curves

TL;DR

This paper extends the canonical-lift construction for ordinary elliptic curves to arbitrary families over p-adic schemes, providing a formal framework that simplifies proofs of related recent results.

Contribution

It generalizes the canonical-lift construction to families over formal schemes, utilizing Witt vectors and moduli space properties.

Findings

Canonical lift extends uniquely to families over p-adic schemes.

Universal ordinary elliptic curve admits a canonical lift.

Facilitates formal proofs of recent elliptic curve results.

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 Erdo\u{g}an.