# The group of $\mathbb{K}$--rational points for isomorphic elliptic curves

The Springer text by Tom Apostol on Dirichlet series and modular forms, which I have, defines modular functions and modular forms on page 34 and on page 114 respectively, not to mention the Springer graduate text by N. Koblitz.

My questions are numbered below, 1, 2, 3.

Let $E_{1}/\mathbb{K}$, $E_{2}/\mathbb{K}$, be two isomorphic elliptic curves over the same field $\mathbb{K}$. Suppose they both are modular.

1. Is it safe to say they have the same group $E(\mathbb{K})$ of $\mathbb{K}$--rational points, up to isomorphism?

2. Do they have the same torsion subgroups, up to isomorphism?

Let $\Lambda$ be their Gaussian lattice on the upper complex plane when \begin{equation} \overline{\mathbb{K}} = \mathbb{C}. \end{equation} By "Gaussian lattice" I mean that lattice on $\mathbb{C}$ for which all the coordinate points ((u, v) = u + vi) are Gaussian integers, \begin{equation} u + vi \in \mathbb{Z}[i], \end{equation} where the ring $\mathbb{Z}[i]$ has a Euclidean valuation. Consider the two curves $E_{1}/\mathbb{\Lambda}$, $E_{2}/\mathbb{\Lambda}$.

$3$. Do the points on the corresponding complex parametrizations on the upper complex plane have the same group transformations from $SL(2, \mathbb{Z})$?

(Manually incorporating an "answer" as an edit.)

Thank you [Bruno Joyal] for the response.

You're right. I checked. I should have written \begin{equation} E_{1}/\mathbb{C}, \: \: E_{2}/\mathbb{C}, \end{equation} and not \begin{equation} E_{1}/\Lambda, \: \: E_{2}/\Lambda, \end{equation} which admittedly is a notation that does not make sense. So actually my question (3) should be rewritten as, "exactly what group acts (up to isomorphism) on the two complex curves \begin{equation} E_{1}/\mathbb{C}, E_{2}/\mathbb{C}? \end{equation} According to the text by J. H. Silverman (Chapter VI, page 158) it is the notation \begin{equation} \mathbb{C}/\Lambda, \end{equation} that denotes a complex Lie group, where $\Lambda$ is an integer lattice on the upper part of $\mathbb{C}$. But the Silverman text mentions two complex Lie groups, namely $E(\mathbb{C})$ as well as $\mathbb{C}/\Lambda$. So I was wondering \emph{which group} on $\mathbb{C}$ would act on the two correspondingly isomorphic, complex parametrized curves $E_{1}/\mathbb{C}$ and $E_{2}/\mathbb{C}$, which would be the complex parametrizations for $E_{1}/\mathbb{K}$ and $E_{2}/\mathbb{K}$ which are isomorphic in $\mathbb{K}$. Does the modular group act on $E_{1}/\mathbb{C}$ and $E_{2}/\mathbb{C}$?

I trust and I hope I am expressing this question clearly enough for experts in this area. The arithmetic/algebraic geometry of elliptic curves is a very tough area of mathematics IMHO.

If two curves are isomorphic then they have the same groups of points. The map $E\mapsto E(K)$ is a functor, and functors preserve isomorphisms. For the same reason they also have the same torsion subgroups.
I do not really know what you mean by $E/\Lambda$. Are you thinking of period lattices? The notation $E/\Lambda$ does not really make sense.