We extend the Serre-Tate theory of canonical lifts of ordinary abelian varieties to arbitrary families of ordinary abelian varieties parameterised by a p-adic formal scheme S. We show that the canonical lift is the unique lift to W (S) which admits a δ-structure in the sense of Joyal and Buium. ContentsIntroduction 1 1. δ-structures and Witt vectors 2 2. Further properties of δ-structures and Witt vectors 9 3. Group schemes, torsors and extensions with δ-structures 18 4. Canonical lifts of ordinary p-groups 23 5. Canonical lifts of ordinary abelian schemes 28 References 31

Abstract. We show that the canonical-lift construction for ordinary elliptic curves over perfect fields of characteristic p ą 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 Erdogan. 1.Fix a prime number p. Let W denote the usual, p-typical Witt vector functor. Let R be a ring in which p is nilpotent, and write S " Spec R. Let W n pSq denote Spec W n pRq, and let W pSq denote the direct limit colim n W n pSq. We take this limit in the category of sheaves of sets on the category of affine schemes with respect to theétale topology. One could say that W pSq is the correct version Spec W pRq, a construction which, as we discuss below, does not have good properties.We say an elliptic curve E over S is ordinary when all fibers of E, necessarily over points of residue characteristic p, are ordinary. For any morphism f : S 1 Ñ S, we will write E S 1 , or f˚pEq, for the base change S 1ˆS E regarded as an elliptic curve over S 1 in the evident way.The purpose of this paper is to prove the following:Theorem. There is a unique way of lifting ordinary elliptic curves E over affine schemes S on which p is nilpotent to elliptic curves r E over W pSq which is compatible with base change in S and has the property that r E admits a Frobenius lift ψ : r E Ñ F˚p r Eq, where F is the usual Witt vector Frobenius map F : W pSq Ñ W pSq.Note that the requirement here that S is affine is only to simplify the exposition. We will remove it below and allow S to be any p-adic formal scheme, or even what we call a p-adic sheaf. See section 4 for the final statement of the theorem and further details.We call r E the canonical lift of E. In the case S " Spec k where k is a perfect field of characteristic p, our canonical lift agrees with the usual one by the remarks in 7.1. Background on sheavesIn this section and the next, we define W pSq, the infinite-length Witt vector construction when S is a scheme, and even when S is more general. The reason there is something to do is that while Spec W n pRq is a well-behaved construction, the naive infinite-length analogue Spec W pRq is not. For instance, some basic geometric facts This work was supported the Australian Research Council.

The aim of this article is to given an extension of the prismatization functor for p-adic formal schemes (whose construction was first sketched by Drinfeld in [Dri20] and then given by Bhatt-Lurie in [BL22b]) to all schemes over Spec(Z). We then prove some basic properties of this extension (algebraicity, flatness for syntomic morphisms, perfectness of cohomology) and show that for smooth schemes over Q this construction recovers (a version of) the filtered de Rham stack. ContentsIntroduction 1. Preliminaries 1.1. Conventions 2. de Rham cohomology over Q 2.1. Reminders on filtrations and the Cartier stack 2.2. The de Rham stack 2.3. Cohomology of X dR 3. Witt vectors 3.1. Reminders 3.2. The map V(1) : V → W 3.3. The ring scheme W 4. The stack Σ Z 4.1. Hodge-Tate elements and ∆ Z 4.2. The stack Σ Z 5. Prismatization over Z 5.1. The ring stacks W E 5.2. Definition and algebraicity of X ∆ Z 5.3. The Hodge-Tate stack X HT 5.4. Presentations of X ∆ Z 5.5. Comparisons and the cohomology of X ∆ Z References

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