First, I apologise if such a question has been asked before. Please feel free to refer me to the previous question, if it answers my current query then I will delete this post.

I am reading the theory of elliptic curves, and since most of the introductory graduate level material seems to center around the discussion of the structure of the set $K$-rational points, $E(K)$ of an elliptic curve $E$, I noticed that there were two ways one could study $E(K)$.

  1. By computing the finite group $E(K)/mE(K)$, and then by the finite-time algorithm described in the descent argument, we'll have the generators of $E(K)$.
  2. We know by the Mordell-Weil Theorem that $E(K) \cong E(K)_{tors} \times \mathbb{Z}^{r(E)}$ where $r(E)$ is called the rank of the elliptic curve $E$. Now by various results in texts such as Silverman's AEC, we have a way of knowing the torsion part, $E(K)_{tors}$. And it is well known that the rank computation is a non trivial task, in fact there is no finite-time algorithm to compute the rank for all elliptic curves.

It seems to me that we can't have a way to compute $E(K)$ no matter how we try to approach the problem (I'm not aware of any other way to approach it other than the two above).

I am now interested in reading the theories behind each of these approaches, the challenges we face while trying to compute $E(K)/mE(K)$ and $r(E)$. And the reason I have made this post is to seek recommendations from you people, texts , articles, lecture notes anything I can look into to systematically(if such a thing is even possible) begin reading the challenges and what progress we have made to remedy these challenges.

In other words, you can say I'm wondering where should one go next once they've become familiar with the material of AEC and are interested in the discussions regarding ranks etc.(see above). Random searches online only seem to lead a number of various articles, papers but I don't know which one I should start from or the background material I need to know for each one.

I have come across an article by Rubin and Silverberg on ranks of elliptic curves and I think I'll begin by first going through that. But it would be lovely if people here could suggest me some other references as well, maybe there is something I should know first before or after reading Rubin's article?

Background: Assume the reader is familiar with the material of Silverman's AEC. Does not any know any Algebraic Geometry except that presented in the first two chapters of AEC.

Thank you.



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