Rational Points – Algorithm for Computing Rational Points if Jacobian Rank is 0

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Is there a general algorithm that can compute in finite time all rational points on any curve of genus $g\geq 2$ whose Jacobian has rank $0$?

If not, for what special cases such algorithm is known? For genus $g=2$, we have Chabauty0 command in Magma, is it always guaranteed to work? For genus $g=3$, natural families to consider are hyperelliptic curves $y^2=P(x)$ for $P$ of degree $7$ or $8$, or Picard curves $y^3=P(x)$ for $P$ quartic. Is there an algorithm for computing rational points on them if rank if the Jacobian is $0$? Is there a corresponding Magma code?

If yes, can you provide a reference? In Cohen's book, I found the following deep theorem Demyanenko-Manin.

Let C be a curve defined over a number field K. Assume that A is a K-simple Abelian variety such that $A^m$ occurs in the decomposition of the Jacobian J of C up to isogeny over K and that
$$
m > \frac{rk(A(K))}{rk(End_K(A))},
$$
where as usual rk denotes the rank. Then C(K) is finite and can be determined explicitly.

Can we derive the result from here with, say, $K={\mathbb Q}$, $m=1$, and $A=J$? Even if yes, is there a better reference, ideally book/paper where the existence of an algorithm for the rank $0$ case is stated explicitly?

Best Answer

There is an algorithm due to Bjorn Poonen (Computing torsion points on curves, Experiment. Math. 10 (2001), no.3, 449–465) that, given a (not necessarily rational) base-point $P_0$ on the curve, finds all (geometric) points $P$ on the curve such that $[P-P_0]$ is torsion in the Jacobian. This solves an even harder problem! (Although, as far as I know, this has been implemented only for curves of genus 2 with a Weierstraß point as base-point.)

Related Solutions

Rational Points on Hyperelliptic Curves – A Detailed Study

By now there is a fairly rich literature on computing the set of rational points on curves of higher genus, see for example my survey paper on "Rational points on curves".

What one can do for your concrete curve is this. First, one can simplify the curve equation (even more than Noam did); your curve is isomorphic to $$y^2 = x^6 - 12 x^5 - 6 x^4 + 82 x^3 + 1461 x^2 - 630 x + 225.$$ Then most methods aiming at finding the set of rational points use the group of rational points on the Jacobian variety of the curve (which by the Mordell-Weil theorem is a finitely generated abelian group). In your case, this group turns out to be free abelian of rank 2. Which is bad news, since the method of choice ("Chabauty's method") only works when this rank is strictly less than the genus. Since (as Noam already mentioned) the defining polynomial has large (even maximal) Galois group, the next method one could try ("Elliptic curve Chabauty") will be infeasible. So despite all the progress that was made in the last couple of decades or so, a curve like yours (with rank $\ge$ genus and no special geometric or arithmetic properties [the smoothness of the discriminant does not help here]) is still likely to resist our efforts.

So what can be done regarding your question? Using explicit generators of the Mordell-Weil group (which takes some computation to provably find), one can fairly efficiently show that there are no rational points beyond the 18 that Noam found up to any given (even quite large) height bound, for example by using the "Mordell-Weil sieve". And if you are interested in integral points, then one can combine the Mordell-Weil sieve with bounds coming from Baker's method (linear forms in logarithms) to prove that you know all integral points.

If desired, I can add details on any or all of the methods mentioned above.

Rational Points on Rank 0 Picard Curves – $y^3 = x^4 + x + 2$

The below Magma code determines the size of $J_C(\mathbb{F}_p)$ for various primes $p$, and finally compute the GCD of their orders, which gives you a bound on the size of $J_C(\mathbb{Q})$. For a discussion of this, see Section 4.1 of this paper. To determine $J_C(\mathbb{F}_p)$, you can just evaluate the numerator of the zeta function of the reduction of $C$ mod $p$ (see e.g., Hindry--Silverman Diophantine Geometry Exercise C.4).

The below code does this for primes up to 30 and tells you that the GCD of $\#J_C(\mathbb{F}_p)$ for these primes is 1, and hence $J_C(\mathbb{Q})$ is just the trivial group. The code is general and can work for any equation of this form of rank 0. If you do have some non-trivial torsion in $J_C(\mathbb{Q})$, then you will need to work harder to determine the rational points.

I will also note that you can run this on the online magma calculator.

P2<x,y,z> := ProjectiveSpace(Rationals(),2);
C := Curve(P2,z*y^3 - (x^4 + x*z^3 + 2*z^4));
bad := {2,3};
bool := false; p := 2;
torsOrders := {@@};
while  p le 30 do
  p :=  NextPrime(p);        
  p := p in bad select NextPrime(p) else p;
  Cp := Curve(Reduction(C,p));
torsOrders := torsOrders join  {@ Evaluate(Numerator(ZetaFunction(Cp)),1) @};
end while;
GCD(torsOrders); //1

Related Question

Algebraic Geometry – Rational Points on Rank 0 Picard Curves for $y^3=x^4+x$