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 $D\_C$?
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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 $D\_{X\_0(N)}$?
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[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. 



Can one bound the Quadratic Points on Curves?
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.


