Probably this is well known to those who know it.


Got an argument and numerical support that over
number fields elliptic curves in minimal models
might have unbounded number of integral points,
the number depending on the degree of the field.


Set $f(x)=x^3+ax+b$ and consider the curve
$E: y^2=f(x)$.


Chose $x\_1 \ldots x\_n$ such that $f(x\_n)$ is
prime and work in $K=\mathbb{Q}[\sqrt{f(x\_1)},\ldots\,\sqrt{f(x\_n)}]$.


$E$ has the obvious $n$ points $(x\_n,\sqrt{f(x\_n)})$.


Experimentally for $f(x)=x^3-x+1$ over
$\mathbb{Q}[\sqrt{7},\sqrt{61},\sqrt{211},\sqrt{337},\sqrt{991}]$ the five points are linearly independent according to sage so the
rank is at least $5$.


Computing the absolute field is not efficient for me.


Over the rationals there is a conjecture relating
the number of integral points to the rank,
is there a similar conjecture for number fields?


Is there an example (with few primes) when in
this construction the points are linearly dependent?


The same argument works for higher genus.



Argument for unboundedness of integral points of elliptic curves over number fields
joro asks: "Over the rationals there is a conjecture relating the number of integral points to the rank, is there a similar conjecture for number fields?"


Yes, the conjecture is that for a given field $K$, on a quasi-minimal Weierstrass equation for $E/K$, the number of $S$-integral points satisfies
$$
 \# E(R\_S) \le C(K)^{1+\#S+\text{rank} E(K)}.
$$
This is known to be a consequence of the $abc$-conjecture for $K$; see [1]. It is also known unconditionally in the weaker form
$$
 \# E(R\_S) \le C(K,\nu(j\_E))^{1+\#S+\text{rank} E(K)},
$$
where $\nu(j\_E)$ is the number of primes $\mathfrak{p}$ of $R\_K$ such that $\text{ord}\_{\mathfrak{p}}(j\_E)<0$; see [2].


[1] M. Hindry, J.H. Silverman, The canonical height and integral points on
elliptic curves, *Invent. Math.* **93** (1988), 419-450.


[2] J.H. Silverman, A quantitative version of Siegel's theorem: Integral
points on elliptic curves and Catalan curves, *J. Reine
Angew. Math.* **378** (1987), 60-100.


