Solomon zeta functions over arithmetic orders

We prove an effective version of Solomon's first conjecture for lattices over orders in finite-dimensional semisimple algebras over nonarchimedean local fields.

We express the quotient of a partial Solomon zeta function by the corresponding maximal-order zeta function as a finite sum whose terms are determined by finite module-theoretic data and weighted by polynomials defined using the Möbius function of finite submodule posets.

The resulting expression is independent of the chosen maximal overorder.

Our proof is purely algebraic and is first formulated for the refined Bushnell--Reiner zeta functions.

As an application, we obtain explicit formulas for the Solomon zeta functions of all lattices over $\mathbb{Z}_p[\mathbb{Z}/p\mathbb{Z}]$, including non-projective lattices.