Can one bound the Quadratic Points on Curves?

Question

Let $C$ be a nonsingular projective curve defined over $\mathbb{Q}$, which does not admit a map of degree 1 or 2 to $\mathbb{P}^1$ or to an elliptic curve. It is then a consequence of Corollary 3 of [1] that $C$ possesses only finitely many quadratic points; i.e., the set

$$\Gamma_C := \left\{p \in C : [\mathbb{Q}(p) : \mathbb{Q}] = 2\right\}$$

is finite. In particular, there exists a bound $D_C$ such that, if $D$ is a squarefree integer satisfying $|D| > D_C$, then $C(\mathbb{Q}(\sqrt{D})) = C(\mathbb{Q})$.

It is natural to ask if there is an algorithm to effectively compute $D_C$, given a model of $C$. Going through the proof of the Harris-Silverman result does not suggest any such algorithm to me (unless the rank of the Jacobian of $C$ is 0).

Given a model of $C$, is there an algorithm to effectively compute $D_C$?

I would also like to ask if $D_C$ is known when $C$ is the modular curve $X_0(N)$; there are only 55 values of $N$ for which $X_0(N)$ does not have finitely many quadratic points (see Theorem 4.9 in [2] for this list). Let $S$ be this set of 55 integers.

Given $N \notin S$, is there an algorithm to effectively compute $D_{X_0(N)}$?

[1]: J. Harris, J.H. Silverman. Bielliptic Curves and Symmetric Products. Proc. Amer. Math. Soc. 112 (1991), 347-356

[2]: F. Bars. On Quadratic Points of Classical Modular Curves. Manuscript, 2012, available from the author's website.

Answer 1

Hi Barinder!

As far as I know there is not an algorithm to do so. See for instance the following paper of Harris and Silverman:

http://www.ams.org/journals/proc/1991-112-02/S0002-9939-1991-1055774-0/

Especially in the proof of Corollary 3 you can really get a clear picture of the argument. Basically if your curve is not hyperelliptic or bielliptic then you find that the symmetric product of $C$ with itself has only finitely many rational points by Faltings' Theorem. Then you have a finite-degree map of sets between the rational points of the symmetric product and your $\Gamma_C$.

If you had a way to compute $D_C$ for any curve, then you'd have something close to an effective version of Faltings for these symmetric product surfaces. Given that Faltings theorem isn't effective for curves I think we're quite far away from that.

For modular curves in particular, well that question is actually on my upcoming agenda. If you'd like to talk about this some time I would probably not require much persuasion.

Answer 2

I was going to point out that the specific result you cite for quadratic points is actually a theorem of Joe Harris and mine (Proc. Amer. Math. Soc. 112 (1991), 347-356), but I see that stankewicz has already given the reference, so I won't repeat the link. What Abramovich and Harris did later (and this is much harder) is generalize the result to points of degree $d$ for all $d\ge2$, although the obvious generalization turns out to be false, one can't merely assume that there are no maps of degree at most $d$ to an elliptic curve or to $\mathbb P^1$. Anyway, as far as I know, all finiteness results of this sort rely on a theorem of Faltings' (generalizing Vojta's proof of the Mordell conjecture) that is highly noneffective. However, for what it's worth, it is possible to give an effective upper bound for $\#D_C$. More generally, one can give effective constants $K_1$ and $K_2$ such that $\#D_C$ has at most $K_1$ points whose height is greater than $K_2$. Unfortunately, we can't get our hands on those putative $K_1$ points of large height, although one suspects that there aren't any such points.