Starting this site

I put this site together mostly so there'd be one place to point people to — an actual home for my research and a running notebook of what I'm thinking about, rather than scattered PDFs and course folders.

The short version of what I work on: elliptic curves that admit extra symmetry in the form of complex multiplication. Take a curve like $y^2 = x^3 + x$. Ordinarily, "multiplying a point by an integer $n$" is the only kind of scaling structure a curve's point group has. But this particular curve also admits multiplication by $i = \sqrt{-1}$: there's a map $[i]$ sending points to points, compatible with the group law, that genuinely behaves like multiplying by $i$. Curves with this extra structure are said to have CM by the Gaussian integers $\mathbb{Z}[i]$.

That extra symmetry has consequences for arithmetic. My honors thesis works out what it does to division polynomials — the polynomials whose roots are the $x$-coordinates of $\pi$-torsion points, for a Gaussian integer $\pi = n + im$. The punchline is that the composite division polynomial $\Psi_\pi = \psi_\pi \psi_{\bar\pi}$ factors cleanly over $\mathbb{Z}[i]$, and an Eisenstein-criterion argument over the Gaussian integers pins down exactly when those factors are irreducible. That, in turn, tells you the Galois group of the field generated by the torsion coordinates: cyclic of order $p - 1$ when $p \equiv 1 \pmod 4$.

A natural next question is what happens if you swap the Gaussian integers for a different ring with extra units — the Eisenstein integers $\mathbb{Z}[\omega]$, where $\omega = e^{2\pi i/3}$. That ring gives curves of the form $y^2 = x^3 + D$ a six-fold symmetry instead of a four-fold one, and the division polynomials behave differently: the roots are closed under multiplication by $\omega$, which forces $\psi_\alpha(x)$ to be a polynomial in $x^3$ rather than $x^2$. Working out the resulting Galois structure — and where the clean "cut the degree in half" argument from the Gaussian case breaks down — is the subject of a paper I just finished. Both are on the research page, with abstracts and PDFs.

I'll use this blog for shorter, less formal notes: things I'm reading, partial arguments I haven't finished, and general updates.