MSc.Thesis Defense:Beyza Mevlüde AMIR

ON TWISTS OF TUPLES OF HYPERELLIPTIC CURVES

Beyza Mevlüde AMIR
Mathematics, MSc. Thesis, 2025

Thesis Jury

 Assoc. Prof. Mohammad Sadek

Assoc. Prof. Kağan Kurşungöz

Assoc. Prof. Mücahit Akbıyık

Date & Time: 11th, June 2025 – 11:00 AM

Place: FENS L063

Keywords :elliptic curves, hyperelliptic curves, rational points, high-rank twists,

quadratic twist

Abstract

We investigate families of hyperelliptic curves whose Jacobians possess positive

Mordell-Weil rank over Q. Given a square-free polynomial f(x) ∈ Q[x] of degree

at least 3 and fixed odd integers m1, m2, m3 ≥ 3, we construct parametric families

of non-square rational values D such that the Jacobians of the twisted curves

C : Dy^2 = f(x) and C_i : y^2 = Dx_mi +a_i for i = 1,2,3, attain positive Mordell-Weil

rank over Q(u,v1,v2,v3). Our approach involves solving a Diophantine system reducible

to finding rational points on certain elliptic curves defined by intersections

of quadratic surfaces. Exploiting explicit rational parametrizations and leveraging

Silverman's specialization theorem, we demonstrate the existence of infinite-order

points on these Jacobians. This yields rational functions D that simultaneously

induce high-rank twists across the considered families of curves.