Curves over every global field violating the local-global principle

TL;DR

This paper presents an algorithm to construct algebraic curves over any global field that violate the local-global principle, and also constructs curves with a specified number of rational points that have points everywhere locally.

Contribution

It introduces an effective method to produce curves over global fields with specific local and global properties, including violations of the local-global principle.

Findings

Algorithm for constructing curves violating the local-global principle

Method to construct curves with a prescribed number of rational points

Curves with points over every completion of the global field

Abstract

There is an algorithm that takes as input a global field k and produces a curve over k violating the local-global principle. Also, given a global field k and a nonnegative integer n, one can effectively construct a curve X over k such that #X(k)=n and X has points over every completion of k.