On the analytic rank of the twin prime elliptic curve y²=x(x-2)(x-p)

2025/08/17 by Kirti Joshi, Joshi, Kirti
Mathematics · Social Sciences · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Historical and Political Studies #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2508.12340

openalex publication_date 2025/08/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let p≥ 7 and suppose (p,p-2) are twin prime numbers, in [Hatley, 2009], the elliptic curve E_p:y²=x(x-2)(x-p) was considered in the context of a conjecture by Jason Beers about the Mordell-Weil ranks of E_p/ℚ. I show that for p≡ 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≡ 1\bmod 8 the rank of E_p is two, is false as stated because E₇₃ has rank zero. In the light of Theorem 4.1.1, Beers' conjecture needs to be modified: if p≡ 1\bmod 8 then the rank of E_p is zero or two (Conjecture 5.3.1).

Citations

• On the Rank of the Elliptic Curve y²=x(x-p)(x-2)

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