Equivariant Birch-Swinnerton-Dyer conjecture for the base change of elliptic curves: An example

TL;DR

This paper proposes a method to verify the equivariant Birch-Swinnerton-Dyer conjecture for elliptic curves over number fields and demonstrates its application on a specific example involving an S_3-extension.

Contribution

The paper introduces a new approach to verify the equivariant BSD conjecture for elliptic curves over Galois extensions, with explicit verification for a conductor 11 curve.

Findings

Verification of the conjecture for a specific elliptic curve of conductor 11

Application of the approach to an S_3 Galois extension of b2b1

Confirmation of the conjecture in this explicit example

Abstract

Let E be an elliptic curved defined over $\Q$ and let $K/\Q$ be a finite Galois extension with Galois group G. The equivariant Birch-Swinnerton-Dyer conjecture for $h^1(E\times_{\Q} K)(1)$ viewed as a motive over $\Q$ with coefficients in $\Q[G]$ relates the twisted L-values associated with E with the arithmetic invariants of the same. In this paper we prescribe an approach to verify this conjecture for a given data. Using this approach, we verify the conjecture for an elliptic curve of conductor 11 and an S_3-extension of $\Q$.