# Possible counterexample to a theorem assuming Lang's conjecture

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

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. ). 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

• Of course, if this is a counterexample to Theorem 1.3, then the most likely explanation is that there's an error in the proof of Theorem 1.3 and the condition $n\ge8$ needs to be replaced with $n\ge9$ or $n\ge10$. It's much less likely that this would contradict the Bombieri-Lang conjecture. But still, you've found a nice example. (Hmmm... Another possibility is that Theorem 1.3 is misstated and actually only holds on a Zariski open subset of the parameter space, which would allow for a finite number of families of exceptions.) Apr 27, 2014 at 15:00

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.
• Thank you. So adding $\alpha_i^2\ne\alpha_j^2$ probably makes the paper correct?