# Elliptic Curves and Torsion Points

So we know many useful theorems that help characterize torsion points on elliptic curves over $\mathbb{Q}$ such as the Nagell–Lutz theorem which provides a useful way to find torsion points on $E/\mathbb{Q}$ and Mazur's theorem which characterizes the torsion subgroup of $E(\mathbb{Q})$.

However, does there exist any elliptic curve over $\mathbb{Q}$ with no torsion points other than the point at infinity? Are there any examples of such curves?

Thanks

• Curves without any non-trivial torsion have density one in the set of all elliptic curves over $\mathbb{Q}$. You can find many examples of such curves in Cremona's database. – Zev Apr 25 '12 at 6:33

Probably the set of $E/\mathbb{Q}$ with $E(\mathbb{Q})=\{O\}$ has density 1/2 (in an suitable way of ordering curves). A recent (and very deep) result of Manjul Bhargava and Arul Shankar proves that this set has positive density, so there are provably a lot of curves with the property that you request. There's a nice overview of the Bhargava-Shankar result, with some of the history, by Bjorn Poonen. It's freely available at
• Ah, I always read these things too fast. I thought he was asking for a curve that had no rational points at all, other than the point at infinity. So after ruling out nontrivial torsion, you still need to get rank 0. This alternative reading is an interesting question. Simply proving (say) that there are infinitely many curves with $E(\mathbb{Q})_{tors}=O$ is an elementary exercise via congruence conditions, as Landsburg noted. Or one can be fancier, e.g., if $E$ has no rational 2-torsion, then all but finitely many quadratic twists have no rational torsion. – Joe Silverman Apr 28 '12 at 12:51
The curve $y^2+y=x^3-x$ has order 5 mod 2 and order 7 mod 3. Therefore the torsion injects into both $Z/5Z$ and $Z/7Z$ and hence is trivial. This is the first example in Husemoller's book on Elliptic Curves.