Let $X$ be a curve of genus $g\geq 2$ over a number field $K$. If $\mathrm{rk} \,\mathrm{Jac}\, X$ is less than $g$ there is a $p$-adic method of bounding $\# X(K)$ due to Chabauty and Coleman (see http://www-math.mit.edu/~poonen/papers/chabauty.pdf).

Does this method give an algorithm for computing $X(K)$ (say, their coordinates)?

I agree to assume BSD and finitness of Shafarevich-Tate group.

Trying to transform their proof to an algorithm, I met the following bottlenecks. First, we need to compute $\log \overline{\mathrm{Jac}(K)}$ (bar means $p$-adic closure) in order to obtain a $1$-form whose integral will vanish on rational points, which is weaker than computing $\mathrm{Jac}(K)$ but I still do not know any non-conjectural algorithms.

Next, the proof requires choosing base points in each residue class and I do not know how to find them algorithmically. Maybe, a solution can be obtained by extending the base field to where it is easy to find a base point.

Finally, we need an argument for computing zeroes of $p$-adic analytic functions. There is an argument, à la Hensel's lemma, for computing zeroes sign-by-sign, which probably terminates if we search for points over number fields, but I am not sure.


Conjecturally, yes. Check out Section 4.4 of this paper by Nils Bruin and myself.

The point is to combine Chabauty-Coleman with the "Mordell-Weil Sieve". In the following, I will assume for simplicity that the Jacobian of your curve $X$ is simple and that $K = \mathbb Q$. I will also assume that one rational point $P_0$ on $X$ is known, which we use to embed $X$ into $J = \operatorname{Jac} X$.

In any case, assuming finiteness of Sha (say), one can in principle determine the rank $r$ of $J(\mathbb Q)$ and find $r$ independent points. Given this information and a prime $p$ of good reduction, one can determine the space of regular 1-forms $\omega$ over ${\mathbb Q}_p$ on $X$ that kill $J(\mathbb Q)$ under the Chabauty-Coleman pairing to any desired $p$-adic precision. In particular, one can compute the image of this space in the 1-forms mod $p$. If $p > 2$ and there is such a 1-form mod $p$ that does not vanish at any ${\mathbb F}_p$-point of $X$ (which heuristically happens for a set of primes $p$ of positive density), then one can show that each residue class mod $p$ can contain at most one rational point. So we now just have to decide, for each residue class, whether there is a rational point or not. If there is a point, then we will eventually find it by searching for it. If there is no point, then (again heuristically) we should be able to prove this using the Mordell-Weil Sieve as explained in the paper. Basically, one tries to use information obtained from reduction modulo a bunch of further primes $q$, together with the knowledge of a finite-index subgroup of $J(\mathbb Q)$, to get a contradiction.

So when $X$ and $J$ satisfy

  • finiteness of Sha (or BSD)
  • there exist $p$ of good reduction and a "good" differential on $X$ mod $p$ that does not vanish in $X(\mathbb F_p)$
  • the "main" conjecture of this paper

then the procedure outlined above will terminate and determine $X(\mathbb Q)$. Also, if the procedure terminates, then it will produce the correct result. So for practical purposes, we can just use it and hope that it finishes. I have implemented it for curves of genus 2 (with Jacobian of rank 1) over $\mathbb Q$ in Magma. It works very well in practice, even though there is no proof so far that it will always work.

  • $\begingroup$ Thank you! So, the method with zeroes of analytic function can not be made into an algorithm? Also, could you please give a reference on how finiteness of Sha implies computability of $J(\mathbb{Q})$, or I overlooked it in your paper? $\endgroup$ – SashaP Sep 12 '16 at 13:18
  • $\begingroup$ @Sasha 1) The problem with zeros of analytic functions is that you may not be able to detect multiple zeros with a finite amout of computation; this is why we pick a special kind of prime, for which there are only simple zeros. Also, you usually have zeros that do not come from rational points, so you need some additional method to show this (this is the role of the Mordell-Weil Sieve). [to be continued] $\endgroup$ – Michael Stoll Sep 12 '16 at 15:07
  • $\begingroup$ 2) That finiteness of Sha implies computability of $J(\mathbb Q)$ is a standard fact. At least in principle, one can compute the $n$-Selmer group $S_n(J)$ of $J$ for any $n \ge 2$. It contains an isomorphic image of $J(\mathbb Q)/n J(\mathbb Q)$ and the quotient is the $n$-torsion of Sha. So when Sha is finite, then for all sufficiently large prime $n$, $\#S_n(J) = n^r$, where $r$ is the rank of $J(\mathbb Q)$. On the other hand, one will find $r$ independent points in $J(\mathbb Q)$ by systematic search. ... $\endgroup$ – Michael Stoll Sep 12 '16 at 15:11
  • $\begingroup$ ... In practice, this is feasible for hyperelliptic curves with $n = 2$ and in a few other rather special cases. (For elliptic curves, it looks a bit better; if the coefficients of the defining equation are not too large, one can do $n = 2,3,4,8,9$.) $\endgroup$ – Michael Stoll Sep 12 '16 at 15:14
  • 1
    $\begingroup$ Sha: Finding points gives you successively better lower bounds on the rank, computing Selmer group gives you upper bounds. When Sha is finite, the bounds will eventually agree, and then you know that $n$ is sufficiently large. Zeros: The functions are $p$-adic analytic functions, which you can only know to finite precision. How do you want to tell two simple, but very close zeros from a double zero? You would have to distinguish between (say) $t^2$ and $t^2+at^n$ without knowing a bound for $n$. Send me an email if you have further questions. $\endgroup$ – Michael Stoll Sep 12 '16 at 20:26

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.