Torsion Points On Hyperelliptic Jacobian Varieties

This thesis presents a detailed study of explicit methods for constructing hyperellipticcurves over the rationals with new torsion orders on the Jacobian. We mentiontwo methods for this purpose.First, we utilize the relation between hyperelliptic curves and continued fractionsof power series. We find that for any integer N in the interval [3g,4g +1], g ≥3, satisfying specific partition constraints, there exist infinitely many families ofJacobians of hyperelliptic curves of genus g possessing a rational torsion point oforder N. We found some original examples of 1-parameter families of hyperellipticcurves. For example, hyperelliptic curves of genus 3 with the Jacobian possessingtorsion divisor of order 13, genus 4 with order 15, genus 5 with order 17, 18, and 21.In the second part, we present another method to construct hyperelliptic curvesfor which the Jacobians contains a torsion divisor of order quadratic in genus g.For any integer g ≥ 2, we construct hyperelliptic curves of genus g over Q whoseJacobian varieties contain rational torsion points of order N where N = 4g2+2g−2, respectively 4g2+2g−4. These curves introduce previously unobserved quadratictorsion orders and provide new torsion orders. For example, rational torsion pointsin the Jacobians of hyperelliptic curves of genus 4 with torsion order 70, and genus3 with torsion order 20.In the last chapter we work on elliptic curves. It was established which groups canoccur as torsion subgroups of elliptic curves over quartic number fields. Except forsome higher-order groups, we identify the quartic field with the smallest absolutediscriminant such that an elliptic curve over this field has the given torsion.