Number Theory Seminar

The (Dedekind or Hilbert) zeta function of an order $R$ counts ideals $I$ of $R$ with $R/I$ finite. We consider a "high-rank" generalization (known as the Quot or lattice zeta function) that counts submodules of $R^{\oplus n}$ with finite quotient, and an "unbounded-rank" analogue that counts (with certain weight) finite modules over $R$, known as the Coh or Cohen--Lenstra zeta function. I will explain some recent progresses on explicit computations on quadratic orders, with a mix of tools from commutative algebra, $p$-adic integrals, and algebraic combinatorics. It turns out that such computations help discover new Rogers--Ramanujan type identities and their "finitizations".