On the analytic rank of the twin prime elliptic curve $y^2=x(x-2)(x-p)$

TL;DR

This paper investigates the analytic rank of a family of elliptic curves associated with twin primes, providing formulas for their root numbers, confirming some of Beers' conjectures, and disproving others through explicit examples.

Contribution

It derives a formula for the global root number of the elliptic curves $E_p$, confirms lower bounds on their ranks for certain primes, and refines Beers' conjecture based on counterexamples.

Findings

For $p ot\equiv 1 mod 8$, the analytic rank of $E_p$ is at least one.

The global root number formula applies to all twin prime pairs.

Counterexample: $E_{73}$ has rank zero, contradicting Beers' conjecture for $p \\equiv 1 mod 8$.

Abstract

Let $p\geq 7$ and suppose $(p,p-2)$ are twin prime numbers, in [Hatley, 2009], the elliptic curve $E_p:y^2=x(x-2)(x-p)$ was considered in the context of a conjecture by Jason Beers about the Mordell-Weil ranks of $E_p/\mathbb{Q}$. I show that for $p\equiv 3,5\bmod 8$, the analytic rank of $E_p$ is at least one (Theorem 1.1.2) in line with Beers' predictions. This is done by finding a formula (Theorem 4.1.1) for the global root number of $E_p$ for all twin prime pairs. I also show that Beers' conjecture, that for $p\equiv 1\bmod 8$ the rank of $E_p$ is two, is false as stated because $E_{73}$ has rank zero. In the light of Theorem 4.1.1, Beers' conjecture needs to be modified: if $p\equiv 1\bmod 8$ then the rank of $E_p$ is zero or two (Conjecture 5.3.1).