Constructing curves of high rank via composite polynomials

2021/02/03 by Arvind Suresh, Suresh, Arvind
Mathematics · #11G30 #11J95 #14G05 #14G25 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2102.02113

openalex publication_date 2021/02/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let k be a number field. We refine a construction of Mestre--Shioda to construct (infinite) families of hyperelliptic curves X/k having a record number of rational points and record Mordell--Weil rank relative to the genus of g of X. Over k=ℚ, we obtain modest improvements on the current published records, and these improvements become more significant as k gets larger. For example, we obtain curves over the real cyclotomic field k = ℚ(cos(π/(g+1))) having at least 16(g+1) k-points and rank at least 8g. The defining equations for the curves are closely related to the classical Chebyshev polynomials, and in special cases, we recover families studied (for example) by Mestre, Shioda, Brumer, and Tautz--Top--Verberkmoes.

Citations

• Three lectures on elliptic surfaces and curves of high rank

Related

• Elliptic curves of high rank over function fields
• Methods for constructing elliptic and hyperelliptic curves with rational points
• Ranks of rational points of the Jacobian varieties of hyperelliptic curves
• Uniform bounds for the number of rational points on hyperelliptic curves\n of small Mordell-Weil rank
• Explicit Chabauty over Number Fields