Ramification in Division Fields and Sporadic Points on Modular Curves

2018/10/11 by Hanson Smith, Smith, Hanson · 1 citation
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT) #Primary 11G05 #Secondary 14H52

paper · pdf · doi:10.48550/arxiv.1810.04809

openalex publication_date 2018/10/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Consider an elliptic curve E over a number field K. Suppose that E has supersingular reduction at some prime \mathfrakp of K lying above the rational prime p. We completely classify the valuations of the pⁿ-torsion points of E by the valuation of a coefficient of the p^th division polynomial. We apply this description to find the minimum necessary ramification at \mathfrakp in order for E to have a point of exact order pⁿ. Using this bound we show that sporadic points on the modular curve X₁(pⁿ) cannot correspond to supersingular elliptic curves without a canonical subgroup. We generalize our methods to X₁(N) with N composite.

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• Rational isolated j-invariants from X₁(ℓⁿ) and X₀(ℓⁿ)

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