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.
Event Details
January 17, 2023
4:00pm
ESB 4127
, , CA