Low rank specializations of elliptic surfaces

2024/08/05 by Mentzelos Melistas, Melistas, Mentzelos
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Analysis Techniques #FOS: Mathematics #Nonlinear Partial Differential Equations #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2408.02419

openalex publication_date 2024/08/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Let E/ℚ(T) be a non-isotrivial elliptic curve of rank r. A theorem due to Silverman implies that the rank r_t of the specialization E_t/ℚ is at least r for all but finitely many t ∈ ℚ. Moreover, it is conjectured that r_t ≤ r+2, except for a set of density 0. In this article, when E/ℚ(T) has a torsion point of order 2, under an assumption on the discriminant of a Weierstrass equation for E/ℚ(T), we produce an upper bound for r_t that is valid for infinitely many t. We also present two examples of non-isotrivial elliptic curves E/ℚ(T) such that r_t ≤ r+1 for infinitely many t.

Related

• An elliptic surface with infinitely many fibers for which the rank does not jump
• Ranks of elliptic curves over ℚ(T) of small degree in T
• Conjecture: 100% of elliptic surfaces over ℚ have rank zero
• On the typical rank of elliptic curves over \mathbb Q(T)
• Three lectures on elliptic surfaces and curves of high rank