My thesis was accepted

I submitted my honors thesis — On Gaussian Integer Torsion Points of Elliptic Curves — for review, and it's been accepted. That makes the thesis itself officially done. I wanted to write down a fuller version of the work here than the two-paragraph abstract sitting on the research page.

The thesis starts by building up the machinery. Weierstrass equations from Riemann–Roch, the geometric group law, and the analytic picture of an elliptic curve as a complex torus $\mathbb{C}/\Lambda$. That last piece matters more than it looks — it's what makes complex multiplication visible at all. For most lattices $\Lambda$, the only endomorphisms of the torus are multiplication by ordinary integers. But for special lattices, like the one behind $y^2 = x^3 + x$, there's an extra endomorphism: multiplication by $i$. That's the whole definition of CM by $\mathbb{Z}[i]$, and it's easy to lose track of how special it is once you're deep in the algebra of division polynomials.

The heart of the thesis is what that extra symmetry does to torsion. For a Gaussian integer $\pi = n + im$, the composite division polynomial $\Psi_\pi = \psi_\pi \psi_{\bar\pi}$ factors cleanly in $\mathbb{Z}[i][x]$ — that factorization is really just the algebraic shadow of the fact that $[\pi]$ and $[\bar\pi]$ are genuinely different maps once you have $[i]$ available. The harder question is whether those factors are further reducible, and that's where an Eisenstein-criterion argument over the Gaussian integers earns its keep: it pins down irreducibility exactly, which is what lets you conclude the Galois group of the field generated by the $\pi$-torsion coordinates is cyclic of order $p - 1$ when $p \equiv 1 \pmod 4$. It's worth dwelling on why that congruence condition on $p$ is load-bearing — drop it and the irreducibility argument breaks, which is a good gauge of how tightly the whole result depends on the arithmetic of $\mathbb{Z}[i]$.

The thesis closes with where the Eisenstein follow-up paper picks up — swapping $\mathbb{Z}[i]$ for $\mathbb{Z}[\omega]$ and watching the clean quartic story turn into a messier hexic one — since that's the direction I want to keep pushing after this.

None of this happens without Gordan Savin, who advised the thesis. I want to say clearly here what doesn't always fit on a CV: he's the reason this project turned into something real instead of staying a half-formed question about elliptic curves I couldn't quite articulate. Every time I got stuck, he had the patience to let me work through it myself first, and the precision to tell me exactly where my reasoning was soft when I couldn't see it. That combination is rarer than it sounds. I came into my honors degree with enthusiasm and not much else, and the amount I've grown as a mathematician over the past year is really a direct reflection of his mentorship. I'm genuinely grateful for it, and for how much of my own mathematical judgment I've picked up just from watching how he approaches a problem.

Thesis submitted and accepted. Next up: turning the Eisenstein work into something more complete.