Title: Arithmetic on hyperelliptic Jacobians and singular curves
Abstract: Mumford representation encodes each divisor class in the Jacobian of a nonsingular hyperelliptic curve as a unique pair of polynomials. And Cantor’s Algorithm provides an implementation of the addition of two divisor classes in this setting. In my thesis, I extended these ideas to study the arithmetic on Jacobians of singular hyperelliptic curves with geometric genus 0. I introduced a generalized Mumford representation and demonstrate that the addition of two divisor classes can be implemented via Cantor Composition. Furthermore, I constructed an explicit isomorphism between the Jacobians of such curves and the product of additive and multiplicative groups of the base field. This result shows that the exact sequence associated with the partial normalization of curves with geometric genus 0 splits. In contrast, I gave examples of specific families of singular curves with positive geometric genus, for which the exact sequence associated with the normalization does not split.