PhD. Dissertation Defense:Hamide Suluyer

TORSION POINTS ON HYPERELLIPTIC JACOBIAN VARIETIES

 

 

Hamide Suluyer
Mathematics, PhD Dissertation, 2025

 

 

Assoc. Prof. Dr. Mohammad Sadek(Dissertation Supervisor)

 

Asst. Prof. Dr. Ayesha Asloob Qureshi

Asst. Prof. Dr. Ekin Özman

Prof. Dr. Ferruh Özbudak

Assoc. Prof. Dr. Ayberk Zeytin

 

Date & Time: 16th June, 2025 –  02:00 PM

Place: FENS- L063

Keywords: Hyperelliptic Curve, Jacobian, Torsion Order, Continued Fractions,

Modular Curve

 

Abstract

This thesis presents a detailed study of explicit methods for constructing hyperellip-

tic curves over the rationals with new torsion orders on the Jacobian. We mention

two methods for this purpose.

First, we utilize the relation between hyperelliptic curves and continued fractions

of 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 of

Jacobians of hyperelliptic curves of genus g possessing a rational torsion point of

order N . We found some original examples of 1-parameter families of hyperelliptic

curves. For example, hyperelliptic curves of genus 3 with the Jacobian possessing

torsion 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 curves

for 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 whose

Jacobian varieties contain rational torsion points of order N where N = 4g^2 + 2g −

2, respectively 4g^2 +2g −4. These curves introduce previously unobserved quadratic

torsion orders and provide new torsion orders. For example, rational torsion points

in the Jacobians of hyperelliptic curves of genus 4 with torsion order 70, and genus

3 with torsion order 20.

In the last chapter we work on elliptic curves. It was established which groups can

occur as torsion subgroups of elliptic curves over quartic number fields. Except for

some higher-order groups, we identify the quartic field with the smallest absolute

discriminant such that an elliptic curve over this field has the given torsion.