Research

We investigate the algebraic and Galois structures of torsion fields generated by elliptic curves of the form $E: y^2 = x^3 + D$, which admit complex multiplication by the ring of Eisenstein integers $\mathbb{Z}[\omega]$. Unlike the classical quartic symmetry collapse observed in curves with Gaussian multiplication, the presence of a sixth root of unity $\zeta_6$ introduces a hexic symmetry variation in the associated division ideals. We establish recursive relations for the corresponding Eisenstein division polynomials $\psi_\alpha$ for $\alpha \in \mathbb{Z}[\omega]$ and analyze the conditions under which these polynomials yield irreducible Eisenstein configurations over $\mathbb{Q}(\omega)$. Finally, we outline the structural obstructions to achieving direct cyclic transitions in the associated $x$-coordinate splitting fields.

This thesis explores the arithmetic and geometric properties of torsion points on elliptic curves, with a specific focus on curves admitting complex multiplication by the ring of Gaussian integers, $\mathbb{Z}[i]$. We begin by establishing the foundational algebraic geometry of elliptic curves, utilizing the Riemann–Roch theorem to derive the general and short-form Weierstrass equations. After defining the geometric group law and the analytic isomorphism between the complex torus $\mathbb{C}/\Lambda$ and the curve $E(\mathbb{C})$, we provide a rigorous treatment of the torsion subgroup structure.

The core of this work focuses on the construction and factorization of division polynomials. For the specific curve $y^2 = x^3 + x$, we derive explicit multiplication-by-$\pi$ maps for Gaussian integers $\pi = n + im$, and show that the composite division polynomial $\Psi_\pi = \psi_\pi \psi_{\bar\pi}$ factors in $\mathbb{Z}[i][x]$. Applying the Eisenstein criterion over the Gaussian integers, we verify the irreducibility of these factors and conclude that the $p$-torsion subgroup decomposes into distinct eigenspaces when $p \equiv 1 \pmod 4$, and that the Galois group of the field extension of $\mathbb{Q}(i)$ generated by coordinates of $\pi$-torsion points is cyclic of order $p - 1$. These results are supported by computational verification using a custom symbolic algebra implementation in Python.