In the paper "On a problem of Zariski", David Rees presents a counterexample to the following problem of Zariski.

Let $F/k$ be a f.g. field extension, $S$ a f.g. normal integral domain over $k$ such that $Q(S)\supseteq F$. Is $S\cap F$ a finitely generated $k$-algebra?

His construction starts with a smooth curve $C$ in $\mathbb{P}^2$ over $\mathbb{C}$ and $p\in C$, and $\mathfrak{p}=I(p)\subseteq\mathbb{C}[x_0,x_1,x_2]$. It produces a certain ring $A$. Now he shows the implication (of which the details are not important for my question I think):

If $A$ has the Nagata property (defined in the paper), then there exists an $m\in\mathbb{N}$ such that the symbolic power $\mathfrak{p}^{(m)}$ is a principal ideal.

He states that geometrically, this means that there exists a curve $D$ in $\mathbb{P}^2$ which intersects $C$ only in $p$, with multiplicity $m$. Using other results from the paper, it suffices to take a suitable pair $(C,p)$ and show that the ring obtained doesn't have the Nagata property. He does this as follows, and this is where I don't understand a step, since I don't know much about elliptic curves.

Let $C$ be a smooth cubic elliptic curve. If we consider the parametrization of $C$ by elliptic functions, then the condition that there should exist a curve meeting $C$ multiply at $p$ and at no other point is that the value of the parameter at $p$ should be a rational multiple of a period.

It follows therefore that there are points $p$ on $C$ such that no multiple of $p$ is a complete intersection, and for such points the ring constructed above does not have the Nagata property.

Does this property of elliptic curves have a name? What does the "no multiple of $p$ is a complete intersection" part mean? From the text above, this property should guarantee that there are points on a smooth elliptic cubic curve such that for every $m$, there exists *no* other curve in $\mathbb{P}^2$ intersecting $C$ in only this point with multiplicity $m$. I guess we can get rid of $m$ here: there exists $p\in C$ such that there is *no* curve $D$ intersecting $C$ only in $p$. What is this property called (and did I "translate" it correctly?), and where can I read up on it?

Thank you in advance!