Yifeng Huang, USC

Title: Coh zeta functions for quadratic orders

Abstract: The Coh zeta function, a 1/|Aut|-weighted count of a ring's finite modules, is a basic invariant that unifies problems in enumerative algebra such as counting matrices and representations. While these zeta functions for (germs of) singular curves have revealed surprising connections to q-series, making precise predictions has remained challenging. This talk provides a unified picture for the known y2=xn singularities by re-interpreting them as quadratic orders: ramified for odd n and split for even n. This algebraic viewpoint reveals a crucial missing piece in the landscape---the inert quadratic orders---which is necessary to complete the picture (via quadratic twists), and is the focus of the new work.