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.

Name thesis:degree_name

Master of Science

Level thesis:degree_level

masters

Discipline thesis:degree_discipline

Mathematics

Department dc:contributor.department

Mathematics

Grantor dc:publisher

Virginia Tech

Year dc:date.issued

2024

Author dc:creator

• Swanson, Nicolas J. Brennan

Chair dc:contributor.committeechair

• Morrison, Travis William

Committee members dc:contributor.committeemember

• Sikora, Jamie
• Matthews, Gretchen L.
• LeGrow, Jason Travis

• Elliptic curves
• decisional algorithms
• local solubility

Statement dc:rights

• In Copyright

Licence dc:rights.uri

• http://rightsstatements.org/vocab/InC/1.0/

Language dc:language.iso

en

Dc Identifier Other

vt_gsexam:40729

OAI identifier oai:identifier

oai:vtechworks.lib.vt.edu:10919/119085