Have Bushnell and Reiner got it wrong?

Question

Let $M$ be a finite-dimensional simple $\mathbb Q$ algebra and $\Lambda$ an order in $M$. Its zeta-function is defined as $$ Z(s)=\sum_{I}|\Lambda/I|^{-s}, $$ where the sum runs over all left ideals and $s$ is a complex number. In a series of 14 papers in the eighties, Bushnell and Reiner claimed that the sum converges for $Re(s)>1$ and that the zeta function extends meromorphically with a pole at $s=1$. Using completely different methods, I found that in the case of dimension 4 it converges for $Re(s)>2$ only and has a pole at $s=2$. The convergence part, by the way, is easily contradicted in the case $M=M_2({\mathbb Q})$ and $\Lambda= M_2({\mathbb Z})$, the respective algebras of $2\times 2$ matrices, you simply need to restrict to the principal ideals, where already you get a sum which diverges at $s=2$.

Now, am I overlooking a renormalisation here, or have they got it wrong in all these papers? (In the second case, they probably made a mistake in the first paper and copied it along.)

In the second case, is their entire approach damaged or is it a small problem which is easily repaired?

Answer 1

It seems like you are overlooking a renormalisation. To illustrate this point, we ought to consider two cases. I'm sorry for deviating from your notation a bit. We want $A$ to be an algebra containing our order $\Lambda$, and we want $M$ to denote some other module.

First we consider the finitely generated free abelian group $M=\mathbb{Z}^{\oplus n}$ of rank $n$. Then, as a left $\mathbb{Z}$-module, $M$ has Solomon zeta function $$\zeta({}_{\mathbb{Z}}M;s)=\prod_{j=0}^{n-1}\zeta(s-j).$$

Now we consider a maximal $\mathbb{Z}$-order $\Lambda=M_n(\mathbb{Z})$ in the finite-dimensional simple $\mathbb{Q}$-algebra $A=M_n(\mathbb{Q})$. As a left $\Lambda$-module, $\Lambda$ has Solomon zeta function $$\zeta({}_\Lambda \Lambda;s)=\prod_{j=0}^{n-1}\zeta(ns-j).$$

In fact, we may see the formula $$\zeta({}_\Lambda \Lambda;s)=\zeta({}_{\mathbb{Z}}M;ns)$$ directly by Morita theory.

To be as explicit as possible, we identify $M$ with row vectors. Then we have a lattice isomorphism $$\{\text{finite index submodules of ${}_{\mathbb{Z}}M$}\}\to\{\text{finite index submodules of ${}_{\Lambda}\Lambda$}\}$$ that takes a finite index submodule $X$ of ${}_{\mathbb{Z}}M$ to the submodule $Y$ of ${}_\Lambda \Lambda$ whose rows lie in $X$. Under this correspondence, it's straightforward to check that $$|\Lambda/Y|=|M/X|^n.$$

We'll end with a small remark. You spoke about restricting to the principal left ideals of $\Lambda$. In fact, every left ideal of $\Lambda$ is principal. This follows from the fact that $\mathbb{Z}$ is a principal left ideal domain. See, for example, Theorem 17.24 of Lam's Lectures on Modules and Rings.