Conditional algorithmic Mordell

TL;DR

This paper constructs hypothetical Turing machines for computing rational points on hyperbolic curves and certain abelian varieties, with their termination guaranteed by major conjectures in number theory.

Contribution

It formalizes the concept of conditional algorithms for key problems in arithmetic geometry based on unproven conjectures.

Findings

Defines $T_{Mordell}$ for computing $C(K)$ if it terminates.

Defines $T_{Shafarevich}$ for listing abelian varieties with specified properties if it terminates.

Shows that these machines' termination is implied by the Hodge, Tate, and Fontaine-Mazur conjectures.

Abstract

We specify a Turing machine $T_{\text{Mordell}}$ with the following properties. 1. On input $(K,C/K)$, with $K/\mathbb{Q}$ a number field and $C/K$ a smooth projective hyperbolic curve, if $T_{\text{Mordell}}$ terminates, then it outputs $C(K)$. 2. The Hodge, Tate, and Fontaine-Mazur conjectures imply that $T_{\text{Mordell}}$ always terminates. Similarly we specify a Turing machine $T_{\text{Shafarevich}}$ with the following properties. 1. On input $(g, K,S, d)$, with $g, d \in \mathbb{Z}^+$, $K/\mathbb{Q}$ a number field, and $S$ a finite set of places of $K$, if $T_{\text{Shafarevich}}$ 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 $T_{\text{Shafarevich}}$ always terminates.