Numerical Evidence for the Equivariant Birch and Swinnerton-Dyer Conjecture

Abstract

Let $E/ \mathbb{Q}$ be an elliptic curve and $K/ \mathbb{Q}$ a finite Galois extension with group $G$. We write $E_K$ for the base change of $E$ and consider the equivariant Tamagawa number conjecture for the pair $(h^1(E_K )(1), \mathbb{Z}[G])$. This conjecture is an equivariant refinement of the Birch and Swinnerton-Dyer conjecture for $E/K$. For almost all primes $l$, we derive an explicit formulation of the conjecture that makes it amenable to numerical verifications. We use this to provide convincing numerical evidence in favor of the conjecture.