I am dealing with the following problem:
Given a $\mathbb{Q}\left (t\right )$-elliptic curve $E(t)$ with positive rank, prove that for all but finitely many $t_0\in \mathbb{Q}$, the $\mathbb{Q}$-cubic curve that results from replacing $t$ with this particular value $t_0$ is in fact a $\mathbb{Q}$-elliptic curve and has positive rank.
I tried the following: drop the finitely many $t\in \mathbb{Q}$ such that the discriminant of $E(t)$ vanishes. Take a point $P(t)$ of infinite order in $E(\mathbb{Q}(t))$.
Now, suppose we know the following fact, which I shall call "weak Mazur's theorem":
There exists $M>0$ such that every $\mathbb{Q}$-elliptic curve has at most $M$ torsion points.
Then, if I take the $C+1$ points $iP(t)$ where $i=1,…,M+1$, since they are generically different, two of them coincide just for finitely many $t\in \mathbb{Q}$. Therefore, for all but finitely many $t_0\in \mathbb{Q}$, $E(t_0)$ is an elliptic curve and $P(t_0)$ is a point in $E(t_0)$ with no torsion.
So, I was wondering whether there is an easy way to prove that there exists some universal upper bound for the cardinal of the torsion group of $\mathbb{Q}$-elliptic curves (at least not so difficult as the proof of Mazur's theorem).
Alternatively, we may not be able to find that universal bound, but we may be able to find such a bound not for all elliptic curves, but for the ones that arise in the form $E(t_0)$ for a given $\mathbb{Q}(t)$-elliptic curve (which we know to be finitely generated because of Mordell-Weil's theorem for function fields). What do you think?
Best Answer
Let $E$ be an elliptic curve over $\mathbb Q(t)$. Let $P_1(t),\ldots,P_r(t)$ be points in $E(\mathbb Q(t))$ that are independent. Then there is an effective (in principle) constant $C=C(E,P_1,\ldots,P_r)$ such that every number in $$ \bigl\{ t_0\in\mathbb Q : P_1(t_0),\ldots,P_r(t_0)~\text{are dependent} \bigr\} $$ satisfies $h(t_0)\le C$. This follows from a height limit theorem $$ \lim_{h(t_0)\to\infty} \frac{\hat h_{E_{t_0}}(P(t_0))}{h(t_0)} = \hat h_E(P).\qquad (*) $$ (Tate proved an even stronger height limit estimate, but $(*)$ is true for 1-parameter families of abelian varieties.)