# Geometric interpretation of j-invariants of supersingular elliptic curves

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?

• 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 $\mathbb F_{p^2}$. – LSpice May 25 '19 at 20:26
• 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)? – Felipe Lopes May 25 '19 at 20:35
• Well all supersingular elliptic curves are reductions of CM elliptic curves, so in that sense yes, you can. – Will Sawin May 26 '19 at 1:51