Research

Our project starts from a 2013 paper by Andrew Alaniz and Tim Huber, On cubic multisections of Eisenstein series. An Eisenstein series is a power series whose coefficients are divisor sums, and these series are some of the basic building blocks of modular forms. A cubic dissection of a power series keeps only the terms whose exponents fall in one residue class mod 3:

$$\sum_{n \ge 0} a_n q^n \;\longmapsto \sum_{n \equiv r \ (\mathrm{mod}\ 3)} a_n q^n.$$

Alaniz and Huber showed that operations like this act linearly on certain spaces of polynomials in cubic theta functions. That turns questions about the series into linear algebra, and the paper builds several families of integer matrices this way. The first three matrices in one family are

$$\begin{pmatrix} 2 & -1 \\ 1 & 1 \end{pmatrix}, \qquad \begin{pmatrix} 4 & -2 & 1 \\ 4 & 1 & -2 \\ 1 & 1 & 1 \end{pmatrix}, \qquad \begin{pmatrix} 8 & -4 & 2 & -1 \\ 12 & 0 & -3 & 3 \\ 6 & 3 & 0 & -3 \\ 1 & 1 & 1 & 1 \end{pmatrix},$$

and their determinants are $3$, $27$, and $729$. Every determinant Alaniz and Huber computed was a power of three, up to sign. They conjectured that the pattern holds for every matrix in these families, and for this family, that the $n$th determinant is $3^{n(n+1)/2}$.

Our team has been working on that conjecture and on the structure behind it, using ideas from linear algebra and representation theory. I can’t say much more until the preprint is out, but my part has included proofs and the Python code we use to compute and check the matrices.

If you’d like a copy of the preprint once it’s out, email me.