In the classical theory of Complex Multiplication, one considers elliptic curves with an endomorphism ring larger than the integers. In this theory, it's possible to determine the j-invariants of all the distinct classes of elliptic curves of a given discriminant by means of the Hilbert class polynomial.

It is well known that, when the elliptic curve $E$ is defined over a field $K$ of positive characteristic, the endomorphism ring can have rank 4, and the curve is called supersingular. In this case, one can also compute a polynomial, with coefficients in $K$, whose roots are the j-invariants of all distinct classes of supersingular curves.

For example, for elliptic curves over $F_p$, these polynomials have been computed explicitly here. It's a general result that j-invariants of these supersingular curves are in $F_{p^2}$. In fact, from the tables linked, one can see various quadratic factors in the polynomials.

My question is about whether it's possible to attach a geometric structure to these curves and interpret their j-invariants as real values. In the classical CM theory, one can compute $J$ numerically by means of the $q$ expansion. And it's not immediately clear that something analogous can be done outside of characteristic zero.

For example, in the above tables, for p=67, we can find $j^2+8j-22$, whose real roots are $-4\pm2\sqrt{38}$. The problem, of course, is that this polynomial should be solved in $F_{p^2}$, so the square root is defined in terms of a quadratic non-residue of $F_p$.

Given that a common theme in algebraic geometry is to transport results from characteristic zero to positive characteristic, I suspect it's possible to construct a space where the real roots of these equations make sense. What are the references for this theory? If it has not been developed yet, what are the obstructions?

  • $\begingroup$ What do you mean "the square root is defined in terms of a quadratic non-residue"? The square root is what it is, a number with the desired square (defined up to sign); that is the definition. It may live in $\mathbb F_p$, or only in $\mathbb F_{p^2}$. $\endgroup$
    – LSpice
    May 25 '19 at 20:26
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    $\begingroup$ What I mean by that is: the square root function, in non-negative reals, is also the limit of convegence of some numeric procedure. So, in classical CM, if you find, for example, that j(q)=$\sqrt{38}$, and you calculate j(q) by means of the q-expansion, you can be sure it will converge to $\sqrt{38}$. However, for this field, once you know that j=$\sqrt{38}$ there is no way to approximate it. What I'm looking for is: can I define the elliptic curve on some geometric space, so that $j$ has a series expansion, and it converges to $\sqrt{38}$ (or some algebraic value corresponding to it)? $\endgroup$ May 25 '19 at 20:35
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    $\begingroup$ Well all supersingular elliptic curves are reductions of CM elliptic curves, so in that sense yes, you can. $\endgroup$
    – Will Sawin
    May 26 '19 at 1:51

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