# Canonical lifts from $\mathbb F_q$ and CM-theory

One knows that (ordinary) Jacobians of hyperelliptic curves over a finite field $\mathbb F_q$ (mostly of genus 1 (elliptic curves) and 2) are extensively studied by cryptographers, as a platform for the discrete logarithm problem. A very powerful tool for the study of those abelian varieties is the theory of complex multiplication. But to use this theory, one needs a bridge between the varieties over $\mathbb F_q$ and those over $\mathbb C$. This is the theory of Serre-Tate of reductions and canonical lifts.

Even though this correspondance is used all over the place in the literature of hyperelliptic cryptography, it is always done in a fuzzy way without any reference (a magic "we lift $A$ to $\mathbb C$" and the problem is swept under the rug, suddenly everything lifts and reduces nicely :-) ). Is there any reference where this is nicely done ?

I guess what people mean by "we lift $A$ to $\mathbb C$" is taking the canonical lift to $W(\mathbb F_q)$ and injecting this into $\mathbb C$. Then we obtain an abelian variety $\mathbb C^n/\Lambda(\mathfrak a)$ with same endomorphism ring as $A$, and CM-type $(K, \{\varphi_1,...,\varphi_n\})$ (where $\mathfrak a$ is an ideal of $\mathrm{End}(A) = \mathcal O$ embedded in $K$, and $\Lambda$ is given by the CM-type). But then does any other curve $\mathbb C^n/\Lambda(\mathfrak b)$ with $\mathfrak b$ an ideal of $\mathcal O$, reduces well to $\mathbb F_q$ ? In a way that is compatible with the reduction of $\mathbb C^n/\Lambda(\mathfrak a)$ ?

Is there a nice, well-defined, one-to-one correspondance between abelian varieties over $\mathbb F_q$ of endomorphism ring $\mathcal O$ (and isogenies between them) and abelian varieties of the form $\mathbb C^n/\Lambda(\mathfrak b)$ with $\mathfrak b$ ideal of $\mathcal O$ (and isogenies between them) ?

(and since we work with Jacobians, we are also concerned about how those reductions/lifts preserve principal polarizations...)

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I have two references in mind: 1) Messing's book shows how if $A_{/k}$ is an ordinary abelian variety then the Serre-Tate Canonical lift $\mathcal{A}_{W(k)}$ admits an isomorphism $End_k(A) \cong End_{W(k)}(\mathcal{A})$. 2) Deligne's Inventiones paper "Varieties Abeliennes Ordinaires..." shows how the choice of embedding $W(k) \hookrightarrow \mathbf{C}$ doesn't change anything. He therefore shows an equivalence of categories between ordinary abelian varieties over a finite field vs $\mathbf{C}$ whose details escape me at the moment but the paper is 3 pages so you should read it anyway :) – stankewicz Apr 10 '14 at 21:33
Thanks ! I don't think I understand how exactly there is an equivalence of categories between ordinary abelian varieties over a finite field and over $\mathbb C$ in Deligne's paper though... I would expect a result like, fixing an embedding $W(k) \rightarrow \mathbb C$, every variety over $k$ of fixed endomorphism ring $\mathcal O$ lifts to a curve $\mathbb C^n/\Lambda(\mathfrak a)$ (up to isomorphism), and reciprocally every such complex curve reduces well (reduction by a prime over $p$ in a number field $L$ ?) back to $k$. (and isogenies would also lift and reduce well...). – Calodeon Apr 11 '14 at 0:17

As pointed out by Stankewicz (this is just a repeat of their answer) there is an equivalence of categories between ordinary abelian varieties over $\mathbf{F}_q$ and lattices $\Lambda$ with an F; the precise form is Theorem 7 of the paper "Vari\'et\'es ab\'e1iennes ordinaires sur un corps fini", Inventiones mathematicae 1969, Volume 8, Issue 3, pp 238-243 and proved in detail in that paper. I suggest reading the paper: it is marvelous!
Now the proof goes through lifts of abelian varieties, so actually if you read the paper, then your question will be answered. On the other hand, if you just want to use the results, then you can use the proposition on page 242 to see what the abelian variety is in terms of $(\Lambda, F)$.
Finally, if you want to understand polarizations then you have to understand how to go from a datum $(\Lambda, F)$ to the datum corresponding to the dual abelian variety. The answer (implicit in the paper -- see formula (3.2)) is to take dual lattice endowed with the contragredient of $V$ as its $F$.