🔢 mathematics

Solomon zeta functions over arithmetic orders

This paper establishes an effective, purely algebraic proof of Solomon's first conjecture for lattices over orders in semisimple algebras over nonarchimedean local fields by expressing the quotient of partial Solomon zeta functions as a finite sum involving Möbius-weighted polynomials, thereby deriving explicit formulas for all lattices over $\mathbb{Z}_p[\mathbb{Z}/p\mathbb{Z}]$ .

Original authors: Sean B. Lynch

Published 2026-07-15

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Sean B. Lynch

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a master librarian trying to count every single book in a massive, infinite library. But here's the twist: the library is built on a strange, multi-layered floor plan where some shelves are perfectly organized (the "maximal orders"), while others are a bit messy and incomplete (the "non-maximal orders").

For decades, mathematicians have been trying to figure out a special counting tool called a Solomon zeta function. Think of this function as a magical calculator that tells you exactly how many ways you can find smaller, finite collections of books (submodules) hidden inside a specific section of the library (a lattice).

The Problem with the Old Map
Back in the day, a mathematician named Hey figured out how to use this calculator for the perfectly organized shelves. It was easy! But when people tried to use it for the messy, incomplete shelves, they hit a wall.

In 1979, a famous conjecture (a big mathematical guess) by Solomon suggested that the answer for the messy shelves was actually just a fancy version of the answer for the clean shelves. Specifically, if you took the messy answer and divided it by the clean answer, you should get a neat, finite list of numbers (a polynomial).

Two mathematicians, Bushnell and Reiner, proved this guess was true in the 1980s. But there was a catch: their proof was like a magic trick that showed the result existed but didn't tell you how to do the trick yourself. They couldn't give you the actual formula to calculate the numbers. It was like being told, "Yes, the treasure is in the chest," but being handed a map that just says, "Look in the chest," without showing you the key.

The New Key: A Playbook for the Messy Shelves
This paper, written by Sean B. Lynch, finally hands us the key. The author proves an effective version of Solomon's first conjecture. This means he didn't just say the answer exists; he wrote down the exact recipe to find it.

Here is how the new recipe works, using a playful analogy:

The "Shadow" Library: Imagine the messy shelf (your lattice $L$ ) casts a shadow onto the perfect shelf above it (the maximal order $\Lambda'$ ). The author realizes that to count the books on the messy shelf, you first need to look at this shadow.
The "Filter" (The Möbius Function): The messy shelf has some extra clutter that doesn't belong. To clean it up, the author uses a special mathematical filter called the Möbius function. Think of this as a sieve that sifts through the shadow, keeping only the parts that match the specific pattern you are looking for and throwing away the noise.
The Finite List: The magic of this new formula is that it breaks the infinite problem down into a finite sum. Instead of checking an infinite number of possibilities, you only have to check a specific, limited number of "module-theoretic data" points. It's like realizing that even though the library is infinite, the specific pattern you are looking for only appears in a handful of specific, countable spots.

What This Formula Actually Does
The paper provides a way to calculate the ratio between the messy shelf's count and the clean shelf's count.

The Formula: It says the messy count divided by the clean count equals a sum of terms.
The Terms: Each term in the sum is determined by looking at a finite group of items (a finite module) and applying the Möbius filter to it.
The Result: The final answer is a polynomial (a neat list of numbers), just like Solomon guessed. But now, we know exactly how to build that polynomial using only finite, computable steps.

A Real-World Test: The Group Algebra
To prove this new method works, the author applies it to a specific, tricky case: lattices over the ring $\mathbb{Z}_p[\mathbb{Z}/p\mathbb{Z}]$ . This is a mathematical structure involving prime numbers and roots of unity.

Previously, mathematicians could only calculate the counts for "projective" lattices (the nice, well-behaved ones). They were stuck on the "non-projective" ones (the messy, weird ones).

The Breakthrough: Using the new formula, the author successfully calculates the Solomon zeta functions for all lattices in this system, including the messy, non-projective ones.
The Formula: The paper gives an explicit formula involving Gaussian binomial coefficients (a special way of counting subspaces) and powers of the prime number $p$ . It recovers all the old known answers and provides brand new formulas for the cases that were previously unsolvable.

What the Paper Does NOT Say
It is important to note what this paper does not do.

It does not solve the "second" conjecture of Solomon (which was already solved by Iyama using different methods).
It does not rely on the old, non-effective proof by Bushnell and Reiner. In fact, the author's proof is "purely algebraic," meaning it uses the structure of the rings and modules directly, rather than the complex "p-adic zeta integrals" used in the past.
It does not claim to solve the problem for every possible mathematical structure in the universe, but rather for lattices over orders in finite-dimensional semisimple algebras over nonarchimedean local fields.

The Bottom Line
This paper turns a "magic trick" into a "cookbook." It takes a mathematical result that was known to be true but impossible to calculate, and turns it into a step-by-step guide that anyone with the right tools can follow. By using the Möbius function as a filter on finite structures, the author proves that the messy, infinite counting problem can always be reduced to a neat, finite polynomial. The result is a powerful new tool that unlocks the ability to count submodules in situations that were previously off-limits.

Technical Summary: Solomon Zeta Functions Over Arithmetic Orders

Problem Statement
The Solomon zeta function of a lattice $L$ over an arithmetic order $\Lambda$ enumerates the finite-index submodules of $L$ . While explicit formulas exist for Solomon zeta functions over maximal orders (Hey's formula), computing these functions over non-maximal orders remains a significant challenge. Previous work by Bushnell and Reiner established Solomon's first conjecture, proving that the quotient of a local Solomon zeta function over a non-maximal order by the corresponding maximal-order zeta function is a Dirichlet polynomial. However, their proof relied on $p$ -adic zeta integrals and a decomposition of Schwartz–Bruhat functions that was not effective; it established the existence of the polynomial but did not provide an independently computable expression for it. Consequently, there was no effective algorithm to compute Solomon zeta functions for general lattices over non-maximal orders.

Methodology
This paper presents a purely algebraic approach to derive an effective version of Solomon's first conjecture. The methodology departs from the analytic $p$ -adic integral techniques of Bushnell and Reiner, returning instead to Solomon's original tool: the Möbius function of the locally finite partially ordered set (poset) of finite-index submodules.

The core technical framework involves:

Refined Bushnell–Reiner Zeta Functions: The author works primarily with these functions, defined over Dirichlet rings, which generalize Solomon zeta functions. This allows for a more abstract algebraic treatment before specializing to the standard zeta functions.
Independence of Maximal Overorders: A crucial step involves proving that the $\Lambda$ -composition factors of simple left $\Lambda'$ -modules (where $\Lambda'$ is a maximal overorder) are independent of the specific choice of $\Lambda'$ . This relies on the structure theory of maximal orders and ensures that the resulting formulas are well-defined regardless of the chosen maximal extension.
Möbius Inversion on Submodule Posets: The author constructs a formula for the quotient of the partial zeta functions by decomposing the sum over submodules. This involves:
- Defining an ideal $I = \{a \in A : \Lambda' a \Lambda' \subseteq \Lambda\}$ .
- Utilizing the Möbius function $\mu$ of the poset of submodules of the finite module $L'/IL'$ (where $L' = \Lambda' L$ ).
- Expressing the quotient as a finite sum weighted by polynomials $Q_{\Lambda'}(W; s)$ , which are defined via Möbius inversion on the submodule poset of $W = \Lambda' X / IL'$ .

Key Contributions and Results
The paper's primary contribution is Theorem 1.1 (and its refined version, Theorem 4.4), which provides an explicit, effective formula for the quotient:
$\frac{\zeta_{\Lambda}(L, M; s)}{\zeta_{\Lambda'}(L'; s)} = \sum_{\substack{X \le_{\Lambda} L \\ X \simeq M \\ IL' \subseteq \Lambda' X}} |L/X|^{-s} Q_{\Lambda'}(\Lambda' X / IL'; s)$
where $Q_{\Lambda'}(W; s) = \sum_{U \le_{\Lambda'} W} \mu(U, W) |W/U|^{-s}$ .

Key features of this result include:

Effectiveness: The formula expresses the quotient as a finite sum of terms determined entirely by finite module-theoretic data (submodules of a finite module) and polynomials defined by the Möbius function. This allows for explicit computation.
Independence: Although the intermediate terms depend on the choice of the maximal overorder $\Lambda'$ , the final weighted sum is independent of this choice.
Explicit Application: The author applies this general framework to the group algebra $\Lambda = \mathbb{Z}_p[\mathbb{Z}/p\mathbb{Z}]$ . By analyzing the structure of lattices over this order (which decomposes into a fiber product), the paper derives Theorem 1.2. This theorem provides explicit formulas for the Solomon zeta functions of all lattices over $\mathbb{Z}_p[\mathbb{Z}/p\mathbb{Z}]$ , including non-projective lattices (specifically $\mathbb{Z}_p$ and $\mathbb{Z}_p[\omega]$ ), which were not accessible via previous methods like Reiner's fiber-product parameterization or Wittmann's Morita equivalence reduction.

Significance
The paper claims significance by providing the first effective, computable expression for the quotient of Solomon zeta functions over non-maximal orders, thereby resolving the non-effectiveness of the original proof of Solomon's first conjecture. By utilizing a purely algebraic approach based on Möbius inversion, the author bypasses the need for non-effective analytic decompositions. The results extend the computability of Solomon zeta functions to non-projective lattices over specific group algebras, filling a gap in the literature where previous methods failed to handle such cases directly. The work establishes a general algebraic mechanism for computing these zeta functions that is independent of the choice of maximal overorder and applicable to orders satisfying certain Artin–Wedderburn conditions.

More like this