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...)