Research

We continue the study, begun in the my honors thesis, of complex multiplication division polynomials and the abelian extensions they generate. There, the curve $E: y^2 = x^3 + x$ and its ring of Gaussian integer endomorphisms $\mathbb{Z}[i]$ were treated in full detail. Here we turn to the family $E: y^2 = x^3 + D$, $D \in \mathbb{Z} \setminus \{0\}$, which admits complex multiplication by the Eisenstein integers $\mathbb{Z}[\omega]$, $\omega = e^{2\pi i/3}$, and whose automorphism group $\mathrm{Aut}(E) \cong \mathbb{Z}[\omega]^\times$ is cyclic of order six. We show that for an Eisenstein prime $\pi$ lying over a rational prime $p \equiv 1 \pmod 3$, the associated division polynomial $\psi_\pi$ is Eisenstein at $\pi$ and hence irreducible over $K = \mathbb{Q}(\omega)$, that its roots are permuted by multiplication by $\omega$, and that this hexic symmetry produces a four-step tower $K \subseteq K(x_1^3) \subseteq K(x_1) \subseteq K(y_1)$ of respective degrees $\frac{1}{6}(N(\pi)-1)$, 3, and 2, whose product recovers a cyclic Galois extension $K(y_1)/K$ of degree $N(\pi)-1$. Along the way we isolate a genuine structural difference from the Gaussian case: because the Eisenstein norm form $N(n+m\omega) = n^2-nm+m^2$ is not symmetric under $m \mapsto -m$, the naïve composite division polynomial entangles two distinct primes of different norms, rather than a prime and its conjugate. We give a short, self-contained proof of the intermediate cubic step using elementary Kummer theory, and work out the minimal case $N(\pi)=7$ completely explicitly, including an exact factorization of the composite polynomial, by hand, into its two constituent primes over $\mathbb{Z}[\omega]$. We also give an elementary proof of the automorphism and endomorphism identification, remark on the ramified prime $p=3$, and exhibit a rational-plus-quadratic torsion subgroup $\mathbb{Z}/6\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$ on the model curve $E_1: y^2 = x^3+1$ over $K$.

In my honors thesis the curve $E_1: y^2=x^3+x$ was treated in detail: for a split Gaussian prime $\pi$ with $N(\pi)=p\equiv1\pmod4$, the division polynomial $\psi_\pi$ was shown to be Eisenstein at $\pi$, and the coordinates of a single $\pi$-torsion point were shown to generate a cyclic extension $K(x_1,y_1)/K$ of degree $N(\pi)-1$ over $K=\mathbb{Q}(i)$, with Galois group $(\mathbb{Z}[i]/\pi\mathbb{Z}[i])^\times$. Here we ask how this picture changes when the coefficient $1$ is replaced by an arbitrary integer: $E_a: y^2 = x^3+ax$, $a\in\mathbb{Z}\setminus\{0\}$. We show first that the entire derivation is uniform in $a$: for every $a$ with $\pi\nmid 2a$, the same cyclic group of order $N(\pi)-1$ arises, the only modification to the argument being a weighted-homogeneity computation pinning down the exact dependence of the constant term $\psi_\pi(0)=u_\pi\, a^{(N(\pi)-1)/4}$ on $a$, where $u_\pi\in\mathbb{Z}[i]^\times$ is the unit computed previously. So the abstract Galois group never depends on $a$. What does depend on $a$ is the actual field $K(x_1,y_1)\subset\overline{K}$: we construct an explicit isomorphism $E_1\to E_a$, defined over $K(\sqrt[4]{a})$, that intertwines the $\mathbb{Z}[i]$-actions, and use it to show that $K_a(\pi):=K(x_1,y_1)$ for $E_a$ and $K_1(\pi)$ for $E_1$ agree on a specific subfield of index $d=[K(\sqrt[4]{a}):K]\in\{1,2,4\}$ — a quantity depending only on the class of $a$ in $\mathbb{Q}(i)^\times/(\mathbb{Q}(i)^\times)^4$, not on $\pi$ — and, under a mild disjointness hypothesis, generate together a field of degree $d(N(\pi)-1)$ over $K$. We verify this structure completely explicitly, with no hypothesis needed, for the smallest split prime $\pi=1+2i$ ($N(\pi)=5$), where an elementary computation gives $\psi_{1+2i}^{(a)}(x)=(1+2i)x^2+a$ for every $a$, and we exhibit $a=16$, $a=4$, $a=2$ realizing the three cases $d=1,2,4$ by hand.

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.