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Eichler orders, quotient graphs and random walks

Luis Arenas-Carmona, Marco Godoy

TL;DR

This paper develops a comprehensive framework to compute classifying graphs C_Q(𝒪) of Eichler and maximal orders at a place Q from the corresponding data at a place P. Central to the approach is the neighborhood matrix N_P(𝒪) and the weighted quotient wq_P(𝒪), together with a finite obstruction theory that identifies a finite set of coordinates where N_Q(𝒪) may differ, depending only on P. The authors prove that, under suitable geometric conditions on the P-graph, the entire N_Q(𝒪) can be recovered from N_P(𝒪) and the cusps at Q, effectively transferring local information across places. They also provide explicit computations and recurrences (f_n) for Eichler orders, together with detailed examples at P and Q, highlighting when obstructions vanish (e.g., multiplicity-free levels) and when they may persist due to graph symmetries. Overall, the work generalizes Serre’s CF-graph results to broader genera and offers practical tools for computing quotient graphs and obstruction sets in function-field settings.

Abstract

We study the extent to which the quotient of the Bruhat-Tits tree at one place $Q$, associated to a genus of orders of maximal rank, can be computed from the analogous quotient at a different place $P$. We show that this computation can be carried out, except for a small set of vertices depending on $P$, but not on $Q$. We give some geometrical conditions on the quotient at $P$ that ensure that this exceptional set is empty. This generalizes the formulas from a previous work that allow the computation of the quotient graph at all places, for the genus of maximal orders over the projective line. The methods presented here yield similar results for other genera or other curves.

Eichler orders, quotient graphs and random walks

TL;DR

This paper develops a comprehensive framework to compute classifying graphs C_Q(𝒪) of Eichler and maximal orders at a place Q from the corresponding data at a place P. Central to the approach is the neighborhood matrix N_P(𝒪) and the weighted quotient wq_P(𝒪), together with a finite obstruction theory that identifies a finite set of coordinates where N_Q(𝒪) may differ, depending only on P. The authors prove that, under suitable geometric conditions on the P-graph, the entire N_Q(𝒪) can be recovered from N_P(𝒪) and the cusps at Q, effectively transferring local information across places. They also provide explicit computations and recurrences (f_n) for Eichler orders, together with detailed examples at P and Q, highlighting when obstructions vanish (e.g., multiplicity-free levels) and when they may persist due to graph symmetries. Overall, the work generalizes Serre’s CF-graph results to broader genera and offers practical tools for computing quotient graphs and obstruction sets in function-field settings.

Abstract

We study the extent to which the quotient of the Bruhat-Tits tree at one place , associated to a genus of orders of maximal rank, can be computed from the analogous quotient at a different place . We show that this computation can be carried out, except for a small set of vertices depending on , but not on . We give some geometrical conditions on the quotient at that ensure that this exceptional set is empty. This generalizes the formulas from a previous work that allow the computation of the quotient graph at all places, for the genus of maximal orders over the projective line. The methods presented here yield similar results for other genera or other curves.

Paper Structure

This paper contains 27 sections, 38 theorems, 141 equations, 39 figures, 4 tables.

Table of Contents

  1. Introduction
  2. A few conventions on graphs
  3. Statement of the main results
  4. quotient graphs for orders of maximal rank
  5. Proof of Theorem \ref{['t1']}
  6. On obstruction sets.
  7. Proof of Theorem \ref{['t2']}
  8. Proof of Theorem \ref{['t3']}
  9. Proof of Theorem \ref{['t5']}
  10. Explicit computations of the obstruction
  11. Some explicit quotients at $P$
  12. Proof of Theorem \ref{['t12']}
  13. Some explicit quotient at $Q$
  14. Acknowledgments
  15. Introduction
  16. ...and 12 more sections

Key Result

Theorem 3.1

Let $\mathfrak{R}$ be an order of maximal rank in $\mathfrak{A}$, and let $P$ be a place where $\mathfrak{R}$ is maximal. Set $U=X-\{P\}$ and $\Gamma=\mathfrak{R}(U)^*K^*/K^*$. Let $N$ be the normalizer of the order $\mathfrak{R}(U)$ in the projective general linear group $\mathrm{PGL}_2(K)$. Then e

Figures (39)

  • Figure 1: A fine graph with a loop at $v_1$, a double edge between $v_2$ and $v_3$, and a half edge at $v_4$. In latter pictures, the virtual vertices, which are denoted by $*$, are ignored unless they have valency 1, as it is the case for $w$.
  • Figure 2: One possible WRFQ $G|\mathfrak{t}$ corresponding to a fine quotient $G\backslash\backslash\mathfrak{g}$ isomorphic to the fine graph in Figure \ref{['F2']}. No virtual vertices are depicted except for the leaves at the half edges. If this picture represents a fine quotient of a tree $\mathfrak{t}$ where every vertex has valency $2$, the weights indicate that the only ramified (actual) vertices in the map $\mathfrak{t}^{[1]}\twoheadrightarrow G\backslash\backslash\mathfrak{g}$ are the pre-images of $v_4$, where a single edge in Figure \ref{['F2']} corresponds to an edge having wheight $2$ in this picture.
  • Figure 3: In (A), an S-graph with $5$ cusps according to Serre's description. In (B), a cusp in the WRFQ. Here $q=\sharp\mathbb{F}(P)$.
  • Figure 4: Structure of a cusp in a classifying graph of maximal orders.
  • Figure 5: Two merging rays, as in the proof of Lemma \ref{['l41p']} (A), and the line mentioned in the proof of Theorem \ref{['t1']} (B).
  • ...and 34 more figures

Theorems & Definitions (75)

  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Theorem 3.5
  • Proposition 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 5.1
  • ...and 65 more