Descent methods and torsion on Jacobians of higher genus curves

In this thesis we accomplish four main results related to Jacobians of curves. Firstly, we find a large number of hyperelliptic curves of genus 2, 3 and 4 whose Jacobians have torsion points of large order. The genus 2 case is particularly well-studied in the literature, and we provide a new example of a geometrically simple Jacobian of a genus 2 curve with a point of order 25, an order which was not previously known. For geometrically simple Jacobians of curves of genus 3 and 4, we extend the known orders of points, increasing the largest known order in both cases to 91 and 88, respectively. Secondly, we find an explicit embedding of the Kummer variety of a genus 3 superelliptic curve into projective space. This is a natural extension of the embeddings that are already known for the Kummer varieties of hyperelliptic curves of genus 2 and 3. Thirdly, we classify the genus 2 curves whose Jacobians admit a (4,4)- isogeny. We find an infinite family of genus 2 curves for which the elements of the kernel of the (4,4)-isogeny are defined over the ground field, and make partial progress on classifying the genus 2 curves with this property. We also extend Flynn’s example of a genus 2 curve whose Jacobian admits a (5, 5)-isogeny to infinitely many geometrically nonisomorphic curves. Finally, we extend Schaefer’s algorithm for computing the Selmer group of a Jacobian to carry out a (4, 4)-descent on Jacobians of curves that admit a (4, 4)-isogeny.