ALGEBRAIC MODULI OF REAL ELLIPTIC CURVES

Abstract

We study algebraic moduli of real generalized elliptic curves. For this, one needs to study algebraic families of such curves. The most suitable class of parameter spaces seems to be the class of Nash manifolds. It turns out, however, that real generalized elliptic curves do not have fine Nash moduli. The somewhat more restricted moduli problem of so-called oriented real generalized elliptic curves does have fine Nash moduli. We prove this by explicitly constructing a universal family of oriented real generalized elliptic curves over a Nash manifold. It will follow that real generalized elliptic curves have coarse Nash moduli. In fact, the coarse moduli space is the Nash manifold P 1(R). As a consequence, every real generalized elliptic curve E has a real j-invariant j R (E) ∈ R ∪ {∞}. Let E and F be real generalized elliptic curves. Then j R (E) = j R (F) if and only if E and F are isomorphic as real curves. We also give an explicit formula for the real j-invariant of a real generalized elliptic curve defined by the Weierstrass equation y 2 = x 3 + ax + b.