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I am reading about the Mordell-Weil group of an elliptic curve over a number field using primarily Silverman's AEC. While the book is excellent in discussing materials prior to Chapter 8, I think that the exposition in Chapter 10 could have been better. But I don't know if there is any approachable text that discusses this topic in as much detail as Silverman does. Hence, the need to make this post.

In short, I am looking for algorithms to compute the Mordell-Weil group for elliptic curves over number fields or over just $\mathbb{Q}$.

From what I understand by skimming over Silverman and John Cremona's Algorithms for Modular Elliptic Curves book, there is $2$-descent algorithm (what Silverman and other texts like Husemoller, Knapp explain) that works when the set of rational points $E(\mathbb{Q})$ has a $2-$torsion point. And the other is 'general descent'. As far as I know, Silverman does not discuss this. But I would really like to read about it.

Could someone suggest some references for it? I know about Cremona's text (but I guess it discusses it in some special case? I'm not sure so correct me if I'm wrong) and Birch and Swinnerton-Dyer's Notes on Elliptic Curves Volume 1. Could someone tell me what are their prerequisites or are they somewhat self-contained? Also, are there any other references anyone would like to suggest?

Thank you very much.

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    $\begingroup$ Cremona describes 2-descent for both the special case where there's a rational 2-torsion point and the general case of a curve y^2=P(x) with P irreducible. $\endgroup$
    – Noam D. Elkies
    Jan 6, 2020 at 19:45
  • $\begingroup$ oh, thanks! Could you mention some (or all) of the prerequisites one needs to understand the general descent given in Cremona's text? And how does the exposition given there differs from the one in Birch, Swinnerton-Dyer's Notes? My guess is that Cremona explains BSD's thing in a special case you mentioned above? $\endgroup$
    – Shreya
    Jan 6, 2020 at 19:56

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