Cambridge Journal of Mathematics

Volume 12 (2024)

Number 2

The conjecture of Birch and Swinnerton-Dyer for certain elliptic curves with complex multiplication

Pages: 357 – 415

DOI: https://dx.doi.org/10.4310/CJM.2024.v12.n2.a2

Authors

Ashay Burungale (Dept. of Mathematics, California Institute of Technology, Pasadena, CA; The University of Texas at Austin, Austin, TX, USA)

Matthias Flach (Dept. of Mathematics, California Institute of Technology, Pasadena, CA, USA)

Abstract

Let $E/F$ be an elliptic curve over a number field $F$ with complex multiplication by the ring of integers in an imaginary quadratic field $K$. We give a complete proof of the conjecture of Birch and Swinnerton-Dyer for $E/F$, as well as its equivariant refinement formulated by Gross $\href{https://doi.org/10.1007/978-1-4899-6699-5_14}{[39]}$, under the assumption that $L(E/F, 1) \neq 0$ and that $F(E_{tors})/K$ is abelian. We also prove analogous results for CM abelian varieties $A/K$.

Keywords

elliptic curves, Birch and Swinnerton-Dyer Conjecture, complex multiplication

2010 Mathematics Subject Classification

Primary 11G40. Secondary 11G15, 11R23.

Full Text (PDF format)

Dedicated to John H. Coates

The first-named author was partially supported by the NSF grants DMS 2303864 and 2302064.

Received 30 July 2022

Published 18 July 2024