Deciding if a Genus 1 Curve has a Rational Point

Many sources suggest a folklore procedure to determine if a smooth curve of genus 1 has a rational point. This procedure terminates conditionally on the Tate-Shafarevich conjecture. In this thesis, we provide an exposition for this procedure, making several steps explicit. In some instances, we also provide MAGMA implementations of the subroutines. In particular, we give an algorithm to determine if a smooth, genus 1 curve of arbitrary degree is locally soluble, we compute its Jacobian, and we give an exposition for descent in our context. Additionally, we prove there exists an algorithm to decide if smooth, genus 1 curve has a rational point if and only if there exists an algorithm to compute the Mordeil-Weil group of an elliptic curve.