The Birch and Swinnerton-Dyer conjecture for an elliptic curve over $\mathbb{Q}(\sqrt[4]{5})$

TL;DR

This paper establishes an equivalence of the Birch and Swinnerton-Dyer conjecture for a specific elliptic curve over a quartic field with that for certain genus 2 hyperelliptic curves over rationals, supported by numerical verification.

Contribution

It demonstrates a novel reduction of the BSD conjecture from an elliptic curve over a quartic field to hyperelliptic curves over rationals, including numerical evidence.

Findings

BSD conjecture verified numerically for hyperelliptic curves

Equivalence established between elliptic curve over $Q( oot 4 ot 5)$ and hyperelliptic curves over $Q$

Methods for finding this example are detailed and intricate.

Abstract

In this paper we show the Birch and Swinnerton-Dyer conjecture for a certain elliptic curve over $\mathbb{Q}(\sqrt[4]{5})$ is equivalent to the same conjecture for a certain pair of hyperelliptic curves of genus 2 over $\mathbb{Q}$. We numerically verify the conjecture for these hyperelliptic curves. Moreover, we explain the methods used to find this example, which turned out to be a bit more subtle than expected.