From Silverman's AEC, Chapter 8, Section 7: I am trying to prove Nagell-Lutz theorem using the Theorem 7.1, which is
In this theorem $K$ is a number field. Now if I have an elliptic curve over $\mathbb{Q}$ as $E: Y^2 = x^3+Ax +B$ where $A, B \in \mathbb{Z}$. Suppose $ P \in E(\mathbb{Q})$ is nonzero torsion point of exact order $m$. Then I need to show that $x(P), y(P) \in \mathbb{Z}$.
The book says that for the case $m = p^n$, a prime power and $ p > 2$, since $r_v =0$ the result follows immediately.
I am trying to explain this as follows:
We note that ord$_v(p) = 1$ if $v=p$, $0$ if $v$ is a finite(that is, not $\infty$) prime different from $p$.
In any case, $r_v =0$. So, by part (b) of this theorem we have, ord$_v(x(P)) \geq 0$ and ord$_v(y(P)) \geq 0$ for all (finite)primes $v$. Thus $x(P), y(P) \in \mathbb{Z}_v$ for all primes $v$ (Sorry for the unusual notation for the ring of padic integers). Thus $x(P), y(P) \in \mathbb{Z}$. (Can't seem to recall why this basic fact is true).
I would really appreciate it if someone here could check my proof and help me fill the holes in the book's proof.
Thank you.
If the order of a rational number $x$ with respect to all finite primes is nonnegative, then the denominator of $x$ (in lowest terms) must be 1, since if any prime $q$ were to divide the denominator then the order of $x$ with respect to $q$ would be negative. Hence $x$ is an integer.
I think that's all you're missing, right?