A conjecture by Goldfeld says that half of all elliptic curves have rank zero (i.e. their Mordell-Weil group has finite order.)
Are there any known infinite families of elliptic curves (over $\mathbb{Q}$) with only finitely many rational points?
For example, in Silverman/Tate, there is an computation which shows that $y^2=x^3+x$ has exactly one rational point and $y^2=x^3+4x$ has exactly three rational points (not counting the point at infinity). I'm wondering if there are any known parameterizations giving an infinite number of such curves.
Best Answer
This is Corollary 6.2.1 of Chapter X in Silverman's "The Arithmetic of Elliptic Curves": Let $p$ be a prime such that $p\equiv 7$ or $11\bmod 16$. Then the Mordell-Weil group of $E_p: y^2=x^3+px$ has rank $0$ (moreover, the $2$-torsion of Sha is also trivial!). Hence, $E(\mathbb{Q})$ has only finitely many points.
In fact, in Proposition 6.1 of Silverman (same chapter X), it is shown that the torsion subgroup of $E_p(\mathbb{Q})$ is $\mathbb{Z}/2\mathbb{Z}$, so $E_p(\mathbb{Q})$ has only $2$ points.
The only drawback to this example is that all these curves $E_p$ have constant $j$-invariant $1728$.