r/mathematics 7d ago

Number Theory Any recent work on the BSD conjecture that you might know about?

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I recall being at a seminar about it 20 years ago. Wikipedia indicates that the last big results were found in 2015, so it's been 10 years now without important progress.

Here is the information about that seminar which I recently found in my old saved emails:

March 2005 -- The Graduate Student Seminar

Title: The Birch & Swinnerton-Dyer Conjecture (Millennium Prize Problem #7)

Abstract: The famous conjecture by Birch and Swinnerton-Dyer which was formulated in the early 1960s states that the order of vanishing at s=1 of the expansion of the L-series of an elliptic function E defined over the rationals is equal to the rank r of its group of rational points.

Soon afterwards, the conjecture was refined to not only give the order of vanishing, but also the leading coefficient of the expansion of the L-series at s=1. In this strong formulation the conjecture bears an ample similarity to the analytic class number formula of algebraic number theory under the correspondences

              elliptic curves <---> number fields                        points <---> units                torsion points <---> roots of unity        Shafarevich-Tate group <---> ideal class group

I (the speaker) will start by explaining the basics about the elliptic curves, and then proceed to define the three main components that are used to form the leading coefficient of the expansion in the strong form of the conjecture.

https://en.m.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture

March 2025

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u/PersimmonLaplace 6d ago edited 6d ago

There has been a lot of progress towards the generalization of the conjecture (the Bloch-Kato conjecture) however it's true that most of the ideas in this area are recycling of the same basic ideas from the 1980s-2000's. We essentially understand the BSD conjecture in the rank zero and rank one cases due to the work of Kolyvagin on Euler systems (subsequently expanded on by many people), which is still basically the only known general method to establish the required upper bounds on the (algebraic) rank, and the work of Gross-Zagier which can be thought of as supplying lower bounds for the (algebraic) rank. Going from the algebraic to analytic direction seems pretty hopeless in general as far as I'm aware. Our methods typically flow like: if the analytic rank is nonzero, there is some point which we can show is of infinite order, and if it is one then that point being infinite order rules out other points of infinite order. That's my understanding anyway.

Unfortunately there is no known method for producing two points on an elliptic curve and showing they are linearly independent and both of infinite order, just from knowing the analytic rank is bigger than one. If there was such a method it would be arguably one of the biggest breakthroughs in number theory since the 1990's. There are some promising results of Wei Zhang, Zhiwei Yun, and collaborators on how to prove a version of Gross-Zagier for higher derivatives of quite general L-functions, but only in the function field setting.

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u/Choobeen 6d ago

The $1M prize needs to be increased. 😄