Imaginary quadratic fields F with X₀(15)(F) finite

2024/05/15 by Tim Evink, Evink, Tim
Mathematics · #Advanced Topology and Set Theory #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2405.09337

openalex publication_date 2024/05/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Caraiani and Newton have proven that if F is an imaginary quadratic number field such that X₀(15) has rank 0 over F, then every elliptic curve over F is modular. This paper is concerned with the quadratic fields F=ℚ(√(-p)) for a prime number p. We give explicit conditions on p under which the rank is 0, and prove that these conditions are satisfied for 87,5% of the primes for which the rank is expected to be even based on the parity conjecture. We also show these conditions are satisfied if and only if rank 0 follows from a 4-descent over ℚ on the quadratic twist X₀(15)_-p. To prove this, we perform two consecutive 2-descents and prove this gives rank bounds equivalent to those obtained from a 4-descent using visualisation techniques for Sha[2]. In fact we prove a more general connection between higher descents for elliptic curves which seems interesting in its own right.

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