Geometric meaning of fiber of modular parameterization over a point of an elliptic curve?

Question

Given an elliptic curve $E/\mathbb{Q}$ of conductor $N$, parameterization $\psi : X_0(N) \rightarrow E$, and a point $P \in E$, take the fiber $\psi^{-1}(P)$. Its points, being on $X_0(N)$, correspond to equivalence classes of pairs $(E_x, C_x)$.

Is there a geometric meaning for these pairs in relation to the point $P$? Something about these elliptic curves that has something to do with $P$? (Other than the j-invariant solving some polynomial equation with coefficients depending only on $P$ (and $E$ of course))

Anything special about the $C_x$'s in relation to $P$?

Modular parameterization is fascinating, but I just don't understand where it comes from. $X_0$ is for a whole bunch of elliptic curves. $E$ is a single specific one.What's the connection between points on the modular curve, to a single specific point of $E$?

Answer 1

As Stankewicz explains, although elliptic curves appear in two guises in the modular parameterization $X_0(N) \to E,$ first because $E$ is an elliptic curve, and secondly because $X_0(N)$ parameterizes elliptic curves, it is something of a red herring to think of these two appearances of elliptic curves as having anything to do with one another.

The reason that $X_0(N)$ appears in the problem of describing elliptic curves is because elliptic curves have two dimensional $H^1$, and $X_0(N)$ is the Shimura variety over $\mathbb Q$ associated to the group $GL_2$. Thus, as Stankewicz notes, Shimura curves (which are the Shimura varieties attached to twisted forms of $GL_2$) can equally well give parameterizations of elliptic curves.

Now the way we prove things about $X_0(N)$ (e.g. properties of its Heegner points, as in Pete Clark's answer) is using its moduli interpretation. But there are two things to bear in mind:

First, most (maybe all?) properties of the special points on $X_0(N)$, such as the Heegner points, are special cases of general aspects of the theory of Shimura varieties (so although the proofs use the moduli interpretation, the statements can be formulated in a way that doesn't refer to the moduli-theoretic interpretation, but instead refers to the interpretation of $X_0(N)$ as a Shimura variety).

Second, the transfer of information is always from $X_0(N)$ to $E$. So while Heegner points give certain interesting points on $X_0(N)$ defined over class fields of quadratic imaginary fields, which can be mapped down to $E$ to give interesting points on $E$ defined over such fields, if one takes a random point on $E$ defined over a class field of an imaginary quadratic field and pulls it back to $X_0(N)$, it is not so easy to say what is going on with the preimages in general.

Finally, I think remark (3) in Pete Clark's answer is an interesting one. In the Mazur and Swinnerton-Dyer paper that David Hansen refer's to in his first comment, if I am remembering correctly, they also suggest that the images in $E$ of the critical points of the map from $X_0(N)$ to $E$ that lie on the geodesic arc joining $0$ to $\infty$ in the upper half-plane may be worth studying. As with Birch's suggestion of Weierstrass points, I'm not sure how much has been done on this.

Answer 2

Thinking about $X_0(N)$ as a bunch of enhanced elliptic curves is a red herring as far as a description of the modular parametrization goes. For instance, you can obtain a similar "modular parametrization" of an elliptic curve starting from a Shimura curve parametrizing certain abelian surfaces (see http://www.math.columbia.edu/~szhang/papers/Heegner01.pdf , or for an overview, see chapter 4 of http://www.math.mcgill.ca/darmon/pub/Articles/Research/36.NSF-CBMS/chapter.pdf )

Now of course this doesn't rule out the possibility of saying anything special about $C_x$ in relation to P, but I can't imagine what you could say. After all, the modular parametrization breaks up into $X_0(N) \hookrightarrow J_0(N) \twoheadrightarrow E_f$ where the right map is essentially projection.

Now if you start with specific sorts of enhanced elliptic curves, and look at their images under the modular parametrization, you can create Heegner points, and those are VERY useful.

Answer 3

I don't know anything useful to say geometrically about the elliptic curves appearing in the divisor $\psi^{-1}(P)$. For instance, I don't know how to describe which $j$-invariants of elliptic curves arise in this way or even how many distinct $j$-invariants arise. Arithmetically speaking, one gets a $\mathbb{Q}$-rational divisor on $X_0(N)$ of degree equal to the modular degree (assuming $\psi$ is unramified above $P$), which is maybe the most I can say.

It works better if you go the other way: namely, choosing specific points or divisors on $X_0(N)$ and pushing them forward to $E$ yields interesting information. Two important examples:

1) The image of a cusp on $X_0(N)(\mathbb{Q})$ is a torsion point on $E(\mathbb{Q})$ (Manin-Drinfeld).

2) Starting with a point $(E,C)$ on $X_0(N)$ such that $E$ and $E/C$ both have complex multiplication by the same order in an imaginary quadratic field yields a Heegner point on $E$. A priori this point is defined over a ring class field of some imaginary quadratic field $K$, but one can take the sum of the Galois conjugates to a get a $K$-rational point. This construction has been more useful for the arithmetic of $E$ than anything else in the last $40$ years: for instance, there is a nontorsion Heegner trace iff $E(K)$ has rank one. C.f. work of Gross-Zagier, Kolyvagin and many others.

And a remark:

3) I recall from an old paper of Birch that after mentioning Heegner points, he wonders what would happen if one took the image of a Weierstrass point on $X_0(N)$. To the best of my knowledge, this has not been explored. Although more natural from the perspective of algebraic geometry than Heegner points, much less is known about the arithmetic geometry of the Weierstrass divisor on a modular curve, for instance, the precise number field over which each of the Weierstrass points is defined.

Answer 4

If your elliptic curve $E$ has rank $1$ over $\mathbb{Q}$, then there is a point $P$, a Heegner point, in $E(\mathbb{Q})$ with a special point $x=(E_x,C_x)$ in the fibre of $\psi$, i.e. such that $E_x$ has extra endomorphisms (as described in the other answers).

If instead your point $P$ of infinite order is on a curve $E$ of rank $2$ over $\mathbb{Q}$, then I believe there is nothing known about the points in the fibre. The degree $d$ of $\psi$ is getting large quite quickly. Each of the $d$ points will be defined over a large number field $K_x$. Probably in most cases the points in the fibre will form one single Galois-orbit. I believe this was verified for the generators of the curve of conductor 389. And there did not seem anything special about the points in the fibre.

It is a strange coincidence that elliptic curves over $\mathbb{Q}$ admit a cover by a curve which parametrises elliptic curves with extra structure. Given that there is this coincidence, one can ask what happens if one takes the points $x=(E,C)$ for different $C$'s on $X_0(N)$ and map them to $E$. This point $P=\psi(x)$ will not be defined over $\mathbb{Q}$ unless $N=27$. (There are two cm curves of conductor 27 with a 27 isogeny between them; the point $P$ is torsion in this case.) Otherwise, if the $j$-invariant of $E$ is not in $\tfrac{1}{2}\mathbb{Z}$ then $P$ will be of infinite order, but it will be defined over a number field much larger than $\mathbb{Q}$.