Conditional algorithmic Mordell

2024/08/21 by Levent Alpöge, Alpöge, Levent, Brian Lawrence +1 · 1 citation
Computer Science · #Algebraic Geometry (math.AG) #Computability, Logic, AI Algorithms #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2408.11653

openalex publication_date 2024/08/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We specify a Turing machine T_Mordell with the following properties. 1. On input (K,C/K), with K/ℚ a number field and C/K a smooth projective hyperbolic curve, if T_Mordell terminates, then it outputs C(K). 2. The Hodge, Tate, and Fontaine-Mazur conjectures imply that T_Mordell always terminates. Similarly we specify a Turing machine T_Shafarevich with the following properties. 1. On input (g, K,S, d), with g, d ∈ ℤ⁺, K/ℚ a number field, and S a finite set of places of K, if T_Shafarevich terminates, then it outputs the finitely many polarized g-dimensional abelian varieties A/K, with polarization of degree d, having good reduction outside S. 2. The Hodge, Tate, and Fontaine-Mazur conjectures imply that T_Shafarevich always terminates.

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• Determining the rational points on a curve of genus 2 and Mordell-Weil rank 1

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