Possible counterexample to a theorem assuming Lang's conjecture

Question

Looks like I found a counterexample to a theorem assuming Lang's conjecture, but not sure it is correct.

Boundedness of Mordell–Weil ranks of certain elliptic curves and Lang’s conjecture p. 2

Theorem 1.3. Let $K$ be a finitely generated field over $\mathbf{Q}$. Let $n \ge 8$ be an integer and let $\alpha_i, (i=0,\ldots,n)$ be fixed elements of $K$. Suppose Conjecture 1.2 holds for $k$ a finitely generated field over $\mathbf{Q}$ (cf. [2]). Then there are only finitely many elliptic curves of the form $y^2=a x^4 + b x^2 +c \; (a,b,c \in K)$ which have $\alpha_i$ as the $x$-coordinates of some $K$-rational points. In particular the Mordell–Weil ranks of such elliptic curves are bounded.

Let $K=\mathbb{Q}[\sqrt{21}], a=c=\frac{-\frac{112}{75} s^{4} + \frac{6647}{4725} s^{2} - \frac{83521}{33339600}}{s^{2}} ,b=\frac{\frac{1799}{75} s^{4} - \frac{171377}{37800} s^{2} + \frac{21464897}{533433600}}{s^{2}}, s \in K$.

Let $P(x)=a x^4 + b x^2 + a$. The discriminant depends on $s$ and $P(x)=P(-x)$.

Consider the elliptic curve $y^2=P(x)$ for $s$ for which the discriminant doesn't vanish.

Let $n=9$ and $\alpha_i=\{1,2,4,1/2,1/4,-1,-2,-4,-1/2,-1/4\}$

$$P(1)= \left(21\right) \cdot s^{-2} \cdot (s - \frac{17}{84})^{2} \cdot (s + \frac{17}{84})^{2}$$ $$P(2)= \left(\frac{1764}{25}\right) \cdot s^{-2} \cdot (s^{2} + \frac{289}{7056})^{2}$$ $$P(4)= 289$$ $$P(1/2)= \left(\frac{441}{100}\right) \cdot s^{-2} \cdot (s^{2} + \frac{289}{7056})^{2}$$ $$P(1/4)= 289/256$$

All of the above are squares as are $P(-x)$.

For all infinitely many admissible choices of $s$, $\alpha_i$ are $x$-coordinates.

The $j$ invariant of the Jacobian depends on $s$.

Is this really a counterexample to Theorem 1.3?

$a,b$ in machine readable form:

 a=c=-1/33339600*(289+7056*s^2)^2/s^2+289/189
 b=257/533433600*(289+7056*s^2)^2/s^2-4913/756

Answer 1

Yes, this is a counterexample to Theorem 1.3. But it looks like the issue is with the proof of Theorem 1.3, and is not relevant to Lang's conjecture. Namely, your example contradicts Theorem 4.2 of that paper, which does not rely on Lang's conjecture. Then the proof of Theorem 1.3 relies on Theorem 4.2, in addition to relying on Lang's conjecture. The only proof given for Theorem 4.2 is "by the argument in [4], we obtain the following theorem", so it is not completely surprising if an error occurs there.

Added later: In fact, the problem is that the author needed to assume that $\alpha_i^2\ne\alpha_j^2$ for $i\ne j$, rather than just that $\alpha_i\ne\alpha_j$. Otherwise everything he says about $W_n$ is wrong. Namely, given elements $\alpha_0,\alpha_1,\dots,\alpha_n$ of a number field $K$, he defines $W_n$ to be the subvariety of $\mathbb{P}^n$ defined by the equations $$ \left| \begin{array}{cccc} 1&1&1&1 \\ \alpha_0^2&\alpha_1^2&\alpha_2^2&\alpha_i^2 \\ \alpha_0^4&\alpha_1^4&\alpha_2^4&\alpha_i^4 \\ Y_0 & Y_1 & Y_2&Y_i \end{array} \right| = 0\,\,\,(i=3,4,\dots,n). $$ Theorem 4.2 asserts that if $n\ge 8$ then the only curves on $W_n$ of genus $0$ or $1$ are the lines $$ (Y_0,\dots,Y_9)=\bigl(\pm(s+t\alpha_0^2),\pm(s+t\alpha_1^2),\dots,\pm(s+t\alpha_n)^2\bigr).$$ Of course, your example produces further genus-$0$ curves on $W_9$, and hence contradicts Theorem 4.2. But as I said, the real issue is that the author should have assumed that the $\alpha_i^2$ are pairwise distinct, since otherwise everything he says about $W_n$ is wrong.