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Has there been any progress toward the Birch and Swinnerton-Dyer conjecture after

The current status of the Birch & Swinnerton-Dyer Conjecture

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No, the conjecture is still wide open for rank $r\geq 2$.

The closest thing to progress is the work of Bhargava and Shankar that quantifies the rank $0$ case and shows that BSD holds for a positive proportion of elliptic curves. You can find it here:

For subsequent developments, see for example

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    $\begingroup$ The difference between the $r \geq 2$ case and the $r \leq 1$ case is the lack of the Gross-Zagier formula, which relates the analytic rank and the algebraic (Mordell-Weil) rank of an elliptic curve, which is the essence of the conjecture. $\endgroup$
    – Stanley Yao Xiao
    Commented May 26, 2017 at 13:35
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    $\begingroup$ @StanleyYaoXiao I would say this is not quite correct. I think instead the difficulty in the $r \geq 2$ case is the lack of a construction of a Heegner point, or rather that the Heegner point is torsion. The fundamental difficulty in BSD is probably in finding rational points rather than in proving that they are nontorsion once you've found them. It's similar to the Tate conjecture, Hodge conjecture, etc. where I would say the fundamental difficulty is just finding algebraic cycles. $\endgroup$
    – Will Sawin
    Commented May 29, 2017 at 16:31
  • $\begingroup$ I haven't looked at Bhargava et al's work, do analytic/probabilistic methods seem capable of leading to complete proofs of BSD ? Or even just of almost sure-BSD ? $\endgroup$
    – plm
    Commented Jul 12, 2023 at 7:49
  • $\begingroup$ @WillSawin, thank you. Do you think that BSD follows from a conjunction of standard conjectures on algebraic cycles in a broad sense ? I have to learn about Lichtenbaum et al's formalism, and don't know the type of methods that are expected to be required for his program or the proof of Bloch-Kato's Tamagawa number conjecture, nor the amount of overlap between the 2, eg if one subsumes the other. Any good reference or gut feeling for that ? :) $\endgroup$
    – plm
    Commented Jul 12, 2023 at 8:03
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    $\begingroup$ @plm I don't think purely geometric conjectures about cycles will imply arithmetic conjectures about BSD, but I do think BSD can be expressed in cycle-theoretic language as a special cade of general conjectures about algebraic cycles like the Bloch-Kato conjecture. $\endgroup$
    – Will Sawin
    Commented Jul 12, 2023 at 13:11
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Benedict Gross recently gave a series of lectures here at the University of Virginia on things related to the Birch and Swinnerton-Dyer Conjecture. One of the recent notable developments he mentioned is the work of Yun and Zhang. It is about the function field analogue but they obtain information about the full Taylor series of the $L$-function. The paper has recently been accepted at Annals of Mathematics.

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    $\begingroup$ There's a popular article about Yun and Zhang here quantamagazine.org/… $\endgroup$
    – j.c.
    Commented May 3, 2018 at 20:14
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Searching `Birch and Swinnerton-Dyer conjecture' by Google, there are two short preprints on this theme:Yongxiong Li, Yu Liu, Ye Tian and K.Morita.

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    $\begingroup$ Although the latter has been updated earlier this month, the unfixable error indicated last July at mathoverflow.net/questions/244459/congruent-number-problem remains unchanged. $\endgroup$
    – nfdc23
    Commented May 27, 2017 at 12:31
  • $\begingroup$ I don't know the detail. If you think so, you should send a mail to the author. $\endgroup$
    – user
    Commented May 27, 2017 at 13:16
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    $\begingroup$ No, the same author posted a 2-page "proof" of BSD for all elliptic curves over $\mathbf{Q}$ in 2013 (arxiv.org/pdf/1305.0392v1.pdf), so contacting this person to discuss basic but fatal errors in a later paper on a related theme is likely to repeat the experience of communicating with J.S.R. and s.jonathan at the link in my first comment. A referee at a journal can try to convey errors to this author, if the paper ever gets that far in the review process. I prefer not to discuss this kind of stuff any further; I get enough of it to deal with as a journal editor. $\endgroup$
    – nfdc23
    Commented May 27, 2017 at 14:48

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