May 13, 2024: The (strong) Birch–Swinnerton-Dyer conjecture: an informal introduction (for a general audience)

The focus of the project [K–Stoll 2023] was the conjecture of Birch and Swinnerton-Dyer (BSD for short) for abelian surfaces. Abelian surfaces are two-dimensional abelian varieties, and abelian varieties are higher-dimensional analogues of elliptic curves. An elliptic curve is an algebraic curve that carries a group structure. This means that we can add two points on the curve to get another point on the curve, and this addition has similar properties as the standard addition. Elliptic curves are important in various contexts within mathematics, for example in the proof of Fermat’s Last Theorem or in cryptography.

Using the numbers of points modulo each prime number on an abelian variety $A$ that is defined over the rational numbers, one can construct a certain function, the $L$ -function of $A$ . The BSD conjecture for $A$ proposes a surprising connection between the analytic behavior of the $L$ -function of $A$ and certain “global” invariants of $A$ . These invariants include properties of the group of rational points on $A$ on the one hand and the number of elements of the mysterious Tate–Shafarevich group $\mathrm{Ш}(A)$ of $A$ on the other hand. Since all other quantities that occur in the conjecture can be computed for given $A$ , the conjecture can be expressed as “ $\mathrm{Ш}(A)$ is finite and has the expected number of elements”.

Birch and Swinnerton-Dyer originally formulated their conjecture for elliptic curves. To prove this version is one of the seven “Millennium Problems” of the Clay Foundation.

For general elliptic curves and even more so for higher-dimensional abelian varieties, the conjecture is wide open. It is not even known that $\mathrm{Ш}(A)$ is always finite. For so-called “modular” abelian varieties with additional properties, some parts of the conjecture are known, however, in particular the finiteness of $\mathrm{Ш}(A)$ . Every elliptic curve defined over the rational numbers is modular, and so it was possible to verify the BSD conjecture for many individual elliptic curves. The goal of the project was to obtain the complete verification of the BSD conjecture also for many modular abelian surfaces. Except in cases that can be reduced to elliptic curves, this had not been done so far even for a single abelian surface. For the verification of the conjecture, we determined the size of $\mathrm{Ш}(A)$ .

The algorithms that we developed and the data on $\mathrm{Ш}(A)$ are also useful outside the framework of this project.