Moduli space of isogeny classes of elliptic curves

Question

The modular curve $Y(1)$ classifies isomorphism classes of elliptic curves, namely its $K$-points for any field $\mathbb Q\subseteq K\subseteq \mathbb C$ correspond via the $j$-invariant to $\mathbb C$-isomorphism classes of elliptic curves defined over $K$.

My question is: what if one wants to classify isogeny classes of elliptic curves defined over $K$? Is there an appropriate moduli space for that? Namely, is there an algebraic variety $Y$ whose $K$-points correspond functorially to $\mathbb C$-isogeny classes (or even better $K$-isogeny classes) of elliptic curves defined over $K$?

Answer 1

When we try to classify elliptic curves up to isogeny, then we use the modular curves $X_0(N)/\mathbb{Q}$, whose non-cuspidal $K$-rational points (for some number field $K$) classify triples $(E/K,E'/K,\phi/K)$ of elliptic curves $E$ and $E'$ defined over $K$, together with an isogeny $\phi:E\to E'$ defined over $K$, with cyclic kernel of size $N$. In other words, the non-cuspidal points on $X_0(N)$ classify $N$-isogenies of elliptic curves.

For instance, $X_0(11)$ has three non-cuspidal $\mathbb{Q}$-rational points, that correspond to the only three $11$-isogenies $121A1\to 121A2$, and $121C1\to 121C2$, and $121B1\to 121B2$, where here I am using Cremona's notation to label elliptic curves. The curve $X_0(11)$ by the way is itself elliptic, it is the curve $11A1$ with a model $$y^2+y=x^3-x^2-10x-20.$$ The Mordell-Weil group of $X_0(11)(\mathbb{Q})$ is isomorphic to $\mathbb{Z}/5\mathbb{Z}$, generated by the point $(16,60)$. Two of the $5$-torsion points correspond to cusps, while the other three correspond to the isogenies mentioned above.

Answer 2

Intuitively such a thing cannot exist, even with level structure, since the "forget isomorphism class and remember only isogeny class" map from the usual moduli space ought to be algebraic, but an algebraic map of curves has finite fibers but (over $\mathbb{C}$, say) isogeny classes of non-isomorphic curves are infinite. Of course there is the problem that the original moduli space is only a stack in the first place, but this obstruction should hold even if you add enough level structure (to both spaces).

For a formal proof, note that the automorphism group of an isogeny class of elliptic curve is the same as $$ (\text{End}(E) \otimes \mathbb{Q})^\times $$ (where $E$ is any representative). This is always at least as big as $\mathbb{Q}^\times$. If you add level structure (in the traditional sense), and change the rules so that you only identify isogenous curves when the isogenies to respect the level structure, you can only obtain a quotient of this by a finite group, which will still be non-trivial. I don't know a formal proof that you can't rigidify the moduli problem in some more interesting way (every way I try to come up with turns into studying isomorphism classes of elliptic curves with level structure, again).