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Impacted Buildings for GL(2)

Malors Espinosa, Zander Karaganis

TL;DR

The paper reinterprets order zeta functions, traditionally counting ideals, through the Bruhat-Tits tree of SL(2,$K$) by introducing impacted buildings (a tree with a basin) and layer-generating functions. It proves a central theorem that the principal-ideal zeta function $\zeta_{\mathcal{O}_n}^P(s)$ equals the vertex-generated zeta $\zeta_{\mathscr{B},\mathcal{O}_n}(q^{-s})$, thereby translating arithmetic questions into combinatorial layer counts. The approach yields explicit closed forms for generating functions in the unramified, ramified, and split cases, and connects these to the known polynomials $P$ and the order zeta functions $\zeta_{\mathcal{O}_n}(s)$ from prior work, offering a unified, geometric viewpoint. This framework provides an alternative route to obtaining local factors in orbital integrals and extends the combinatorial handling of zeta functions beyond rank-2 lattices to principal ideals in Bruhat-Tits buildings, with potential applicability to broader groups and local fields.

Abstract

In this paper we define a generating function for buildings of type $\widetilde{A}_1$ (i.e. trees) that are enhanced with a certain filtration structure. We prove that this generating function recovers the zeta function of certain quadratic orders. We do this by studying how the ideals of the orders distribute in the building of $SL(2, K)$.

Impacted Buildings for GL(2)

TL;DR

The paper reinterprets order zeta functions, traditionally counting ideals, through the Bruhat-Tits tree of SL(2,) by introducing impacted buildings (a tree with a basin) and layer-generating functions. It proves a central theorem that the principal-ideal zeta function equals the vertex-generated zeta , thereby translating arithmetic questions into combinatorial layer counts. The approach yields explicit closed forms for generating functions in the unramified, ramified, and split cases, and connects these to the known polynomials and the order zeta functions from prior work, offering a unified, geometric viewpoint. This framework provides an alternative route to obtaining local factors in orbital integrals and extends the combinatorial handling of zeta functions beyond rank-2 lattices to principal ideals in Bruhat-Tits buildings, with potential applicability to broader groups and local fields.

Abstract

In this paper we define a generating function for buildings of type (i.e. trees) that are enhanced with a certain filtration structure. We prove that this generating function recovers the zeta function of certain quadratic orders. We do this by studying how the ideals of the orders distribute in the building of .
Paper Structure (26 sections, 49 theorems, 178 equations, 3 figures)

This paper contains 26 sections, 49 theorems, 178 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The goal of this article
  3. Results and organization of this article
  4. Previous literature
  5. Impacted Buildings and their Generating Functions
  6. The Generating Functions from a Vertex
  7. The Way Out
  8. Combinatorics, Rationality and the Recurrence Relation
  9. Explicit computations of generating functions
  10. The unramified case
  11. The ramified case
  12. The split case
  13. Order polynomials for $\mathop{\mathrm{\mathrm{GL}}}\nolimits(2)$ and their computations via ideal types
  14. The setup and the Zeta Function of Orders
  15. The Traveling map and the Principal Zeta Function
  16. ...and 11 more sections

Key Result

Theorem A

Let $L/K$ be one of our three cases (i.e. ramified, unramified, or split), $n\ge 0$ and ${{\mathscr{B}}}$ the corresponding impacted building (i.e. ramified, unramified, or split, respectively). A vertex $v$ of ${{\mathscr{B}}}$ represents a (principal, rank $2$) ideal of ${{\mathcal{O}}}_n$ if and

Figures (3)

  • Figure 1: Two examples of "topographic" maps of impacted buildings.
  • Figure 2: The impacted buildings in the unramified, ramified, and split case from left to right.
  • Figure 3: The different cases and their possible types.

Theorems & Definitions (123)

  • Definition A
  • Definition B
  • Theorem A
  • Theorem B
  • Definition 1
  • Remark 1
  • Definition 2
  • Definition 3
  • Remark 2
  • Example 1
  • ...and 113 more