How to explicitly compute lifting of points from an elliptic curve to a modular curve?

2011-02-13T10:03:12Z

Say $E$ is an elliptic curve over the rationals, of conductor $N$. There's a covering of $E$ by the modular curve $X_0(N)$, and if you rig it right then you can define this map over $\mathbf{Q}$: there's a map $\pi:X_0(N)\to E$ of algebraic varieties over $\mathbf{Q}$.

Now say I have an explicit $\mathbf{Q}$-point $P\in E(\mathbf{Q})$. Its pre-image in $X_0(N)$ will be a finite set of points, all defined over number fields. Perhaps a bit more conceptually, the pullback of $\pi$ via the map $Spec(\mathbf{Q})\to E$ induced by $P$ is a scheme $Spec(A)$ where $A$ is a finite $\mathbf{Q}$-algebra.

How would one go about actually computing these number fields in an explicit example? (or computing $A$, if you like). One can do computations in Jacobians of modular curves so easily these days using modular symbols, so I would imagine this is easy for the experts.

As an explicit example let's take a non-torsion point $P$ on an elliptic curve of rank two (so one can't "cheat" and do the calculation using Heegner points or cusps)---for example let $P$ be some random non-zero small height element of Mordell-Weil mod torsion in the rank two curve of conductor 389. What number fields do the points in the modular curve that map to $P$ cut out?

Answer by Chris Wuthrich for How to explicitly compute lifting of points from an elliptic curve to a modular curve?

2011-02-13T12:20:56Z

I do not believe that modular symbols will help. They are good to detect that the rank is positive, but since they only produce torsion points in $E$, they are unfortunately of no use to construct points of infinite order. I fear also that they can not give you the fields that you are after.

If you have the modular parametrisation from a model of $X_0(N)$ to $E$, you can do it easily. But this is not what your after, I know.

Here is another idea. Take a prime $p$ and suppose the point $P$ lies in the formal group. If not you could multiply it to lie in it, but you would change the field. Then $P$ is $p$-adically close to $O$. Now compute its image $z=\log_{\hat{E}}(P)$ under the formal logarithm map into the $p$-adic points of the Lie algebra. The formal version of the modular parametrisation from $X_0(N)$ to $E$ is just $q\mapsto\sum \frac{a_n}{n} q^n$, which has a local inverse around $O$. So you will find a value of $q\in p\mathbb{Z}_p$ that maps to $z$. There is a Tate elliptic curve $A$ defined over $\mathbb{Q}_p$ with multiplicative reduction whose $q$-parameter is this $q$. Computing $j(q)$ to sufficiently high precision, it should be possible to guess its minimal polynomial over $\mathbb{Z}$ and then, using this guess one can verify the claim. (But I have not attempted to do this, so I have no idea if it is feasible.)

(edit:) This can be done for $p=\infty$, too. If $P$ is sufficiently close to $O$ in $E(\mathbb{R})$ then one can recover the $\tau$ in the upper half plane on the imaginary axis with largest imaginary part as above. (This corresponds to integrating the modular form along the imaginary axis until we get $P$ for the first time.) Then one can recover the minimal polynomial of $j(\tau)$ by acting on it with the cosets representatives of $\Gamma_0(N)$.

Note also, once that $j(q)$ is known. We also have to check if the corresponding curve has a $N$-isogeny over its field of definition or otherwise one needs to enlarge it.

In any case, your field will have large degree. In fact, I do not see any reason to believe that the degree of this field should be smaller than the degree of the modular parametrisation.

The question is similar to this .

How to explicitly compute lifting of points from an elliptic curve to a modular curve? - MathOverflow

How to explicitly compute lifting of points from an elliptic curve to a modular curve?

Answer by Chris Wuthrich for How to explicitly compute lifting of points from an elliptic curve to a modular curve?