2025/05/19 by Sa’ar Zehavi, Zehavi, Sa'ar
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2505.12947
openalex publication_date 2025/05/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We present a practical, unconditional algorithm for determining the S-integral points on any elliptic moduli problem Y/ℤ[1/S] -- that is, on any geometrically connected curve carrying a non-isotrivial elliptic fibration E → Y. The associated map ΦM\colon Y → M1,1 (the modular period map) plays the role ordinarily filled by a p-adic period map in Chabauty-type methods. Our Modular Chabauty method studies the image and fibres of ΦM, and proceeds in two steps: an Effective Shafarevich step, in which we combine the modularity theorem with Cremona's enumeration of elliptic curves by conductor and list all rational elliptic curves with good reduction outside S; and a Fibre Computation step, in which we compute the S-integral points in the corresponding fibre of ΦM. A Python/Sage implementation computes Y(ℤ[1/S]) for Y=ℙ1∖\0,1,∞\ and for every modular curve Y1(N) with 4≤ N≤ 10 or N=12, for all sets S with ∏p∈ S p2≤ 5⋅ 105, within 3.5 seconds on a standard computer.