Daniel Macías Castillo (ICMAT-CSIC) and David Burns (King’s College London) are the authors of the article “On Refined Conjectures of Birch and Swinnerton-Dyer Type for Hasse-Weil-Artin L-Series,” published in Memoirs of the American Mathematical Society.

Let A be an abelian variety defined over a number field k. By a celebrated theorem of Mordell and Weil, the abelian group that is formed by the set of points of A with coefficients in k is finitely generated. It is also conjectured that the Hasse-Weil L-series L(A,z) of A has a meromorphic continuation to the entire complex plane and satisfies a functional equation with central point  z=1 and, in addition, that the Tate–Shafarevich group of A is finite.

Assuming these conjectures to be true, the Birch and Swinnerton-Dyer conjecture (`BSD’) predicts an explicit formula that relates the leading coefficient in the Taylor expansion of L(A,z) at z=1 to several key algebraic invariants of A, including the order of its Tate-Shafarevich group.

This remarkable prediction is regarded as one of the most important problems in arithmetic geometry. Nevertheless, there are various natural contexts in which it does not encompass the full extent of the interplay between the analytic and algebraic invariants of A. In particular, `refined conjectures of BSD type’ have been formulated and studied by several authors including Bertolini, Darmon, Deligne, Gross, Kato, Kurihara, Mazur, Rotger, Rubin and Tate.

In this article the authors formulate a seemingly definitive refinement of the BSD conjecture for the Hasse-Weil-Artin L-series associated to A and to finite dimensional complex characters of the absolute Galois group of k. They then derive a range of concrete consequences of this conjecture that are amenable to explicit investigation. In particular, they show that their conjecture both refines and extends the existing theory of refined conjectures of BSD type, and also implies the relevant case of the Equivariant Tamagawa Number Conjecture (`ETNC’).

In important special cases they provide strong evidence for the validity of their conjecture. For instance, in the setting of rational elliptic curves and of characters that factor through number fields that are both abelian and tamely ramified, they use the theory of modular symbols and a theorem of Kato to obtain a proof of their conjecture for very general families of such curves and characters. This result also extends the previous verifications of the ETNC in this setting. In certain more difficult settings, they are also able to provide numerical verifications of their conjecture.

To achieve these results, the article carries out an arguably definitive treatment of several topics in the arithmetic of abelian varieties.

Reference

Burns D, Macias Castillo D. On Refined Conjectures of Birch and Swinnerton-Dyer Type for Hasse–Weil–Artin L-Series. Memoirs of the American Mathematical Society. 2024;297(1482). doi:10.1090/memo/1482.