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First Year Report

The main objective of this report is to present a method of finding the two torsion subgroup of the Mordell-Weil group of non-hyperelliptic curves of genus 3 and 4. Example. Let C be the plane quartic C : x3z − x2y2 + xyz2 − y3z − 5z4 ⊂ P2 and let J = J (Q) be it’s Mordell-Weil group. Additionally, assume that we know J ∼= Z/2Z× Z/4Z× Z/8Z or Z/2Z× Z/2Z× Z/4Z× Z/8Z or Z/2Z× Z/4Z× Z/4Z× Z/8Z If we can calculate the two torsion subgroup of J and if J [2] ∼= (Z/2Z), then this is sufficient to conclude that J ∼= Z/2Z× Z/4Z× Z/8Z. Sections 2 and 3 will give the background necessary for our calculations. In section 3, we establish a a bijection between divisors formed from theta hyperplanes to the curves and the two-torsion points, and give a strategy for computing the two torsion subgroup. Sections 4 and 5 describe a method for finding the theta hyperplanes to a non-hyperelliptic genus 3 or 4 curve. Section 6 describes how the two torsion subgroup is calculated using these theta hyperplanes. Finally, Sections 7 and 8 give examples of how the strategy described works in practice.