Trivial torsion subgroup

Question

I am just wondering, suppose we have a curve $y^2 = x^3+ax + b$ defined over $\mathbb{Q}$ and suppose for simplicity $a,b \in \mathbb{Z}$. Can we say something about the torsion subgroup with the only information based on $a,b$? More precisely: can we say for sure that the torsion subgroup is trivial, based on some relation between $a$ and $b$?

Answer 1

Since this is tagged as "Elliptic curves" I'm going to assume you meant $y^2= x^3+ax +b$.

The Nagell-Lutz Theorem gives that all torsion points have integer coordinates-- furthermore the $y$-coordinate of a torsion point necessarily divides the discriminant $D$, given by $-16 (4a^2 +27b^2)$ (in your case, since your elliptic curve is written in a form which implies it has characteristic $\ne 2$ or $3$.) So to compute the torsion subgroup we find all integers whose squares are factors of the discriminant. After you have found every such $y$, find each corresponding $x$-- if $x$ isn't an integer, then we can throw away that point.

After we have found all integer points on the curve whose $y$-coordinates have squares dividing the discriminant, we look to Mazur's theorem, which says that if a point is of finite order, it must have order $n$, where $1 \le n \le 10$ or $n=12$. (This is an incredibly powerful result.) So we compute $nP$ for an integer point $P$, trying the values of $n$ given by Mazur's theorem. If $nP$ has non-integer coordinates for all the candidate $P$, then the torsion subgroup consists only of the point at infinity.

Hope this helped!