From the lattice zeta function to "spiral shifting" operators
January 17, 2023
In 1977, Solomon found a formula for the number of full-rank sublattices of $\mathbb Z^d$ of index n in terms of the Riemann zeta function. The same proof generalizes to counting full-rank submodules of $R^d$, where $R$ is a Dedekind domain. We give a new proof of the case where $R=\mathbb F_q[[T]]$, the power series ring over a finite field, by introducing a fun combinatorial construction, which is some "spiral shifting" operators acting on the set of $d$-tuples of nonnegative integers. I will use concrete examples to demonstrate several nice properties of these operators, hopefully interactively. Time permitting, I will explain its further application to counting submodules of $\mathbb F_q[[T^2,T^3]]^d$ and counting matrices satisfying $A^2=B^3$, $AB=BA$. This talk is based on joint work arxiv: 2210.10215 with Ruofan Jiang.
Part of the motivation of this work originates from algebraic geometry, which I will explain in my AG seminar talk at 4:10-5:10pm, Mon Jan 16, MATH 126. You are encouraged to attend that talk for the other side of the story, but no prerequisite from that talk is required here.