Verification of strong BSD for elliptic curves and abelian surfaces over totally real number fields

• Applicant: Professor Dr. Michael Stoll
• Subject Area: Mathematics
• Term: since 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 441241343

The focus of this project is the conjecture of Birch and Swinnerton-Dyer (BSD for short) for elliptic curves over totally real number fields. 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.A totally real number field is an extension of the field Q of rational numbers that is generated by a root of a polynomial with rational coefficients all of whose roots are real numbers.Let E be an elliptic curve over a number field F.Using the numbers of points on E modulo each prime ideal of F, one can construct a certain function, the L-function of E. The BSD conjecture for E proposes a surprising connection between the analytic behavior of the L-function of E and certain "global" invariants of E. These invariants include properties of the group of F-rational points on E on the one hand and the number of elements of the mysterious Shafarevich-Tate group Sha(E/F) of E on the other hand. Since all other quantities that occur in the conjecture can be computed for a given E, the conjecture can be expressed as "Sha(E/F) is finite and has the expected number of elements".Birch and Swinnerton-Dyer originally formulated their conjecture for elliptic curves over Q. To prove this version is one of the seven "Millennium Problems" of the Clay Foundation.For general elliptic curves, the conjecture is wide open. It is not even known that Sha(E/F) is always finite. For so-called "modular" elliptic curves with additional properties, some parts of the conjecture are known, however, in particular the finiteness of Sha(E/F). Every elliptic curve defined over Q is modular, and so it was possible to verify the BSD conjecture for many individual elliptic curves over Q. In the predecessor of this project, we extended this to some modular abelian surfaces over Q, which are two-dimensional analogues of elliptic curves.The goal of this new project is to obtain the complete verification of the BSD conjecture also for many modular elliptic curves (and, if possible, also abelian surfaces) over totally real number fields F other than Q.The algorithms that we will develop and the data on Sha(E/F) that will result will also be useful outside the framework of this project.

• DFG Programme: Research Grants