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Metacommutation of primes in Eichler orders

In this article, we study the metacommutation problem in locally Eichler orders. From this arises a permutation of the set of locally principal left ideals of a given prime reduced norm. Previous results on the cycle structure were determined for locally maximal orders. As we extend these results, we present an alternative, combinatorial description of the metacommutation permutation as an action on the Bruhat-Tits tree.


Introduction
In this article, we study the metacommutation problem, following work of Conway-Smith [3], Cohn-Kumar [4], Forsyth-Gurev-Shrima [6], and Chari [2]. Conway-Smith [3] first proposed the metacommutation problem in the Hurwitz quaternion order The problem is as follows. If π and ω ∈ O have distinct prime reduced norms nrd(π) = p and nrd(ω) = q, we may factor πω = ω ′ π ′ , with nrd(π ′ ) = p and nrd(ω ′ ) = q, and the choice of π ′ is unique up to left multiplication by units. Define σ ω (π) = π ′ if there is an element ω ′ with nrd(ω ′ ) = ω such that This is a permutation and is well-defined because of the two unique ways to factor πω up to units. The study of this occurrence is called the metacommutation problem.
The metacommutation problem was first studied by Cohn-Kumar [4]. They determined the number of fixed points and the sign of a permutation induced by an element ω ∈ O and partial results on the cycle structure. Forsyth-Gurev-Shrima [6] then viewed the problem as an action of matrix groups on projective space P 1 (F p ) and showed that in the Hurwitz order, all cycles that are not fixed points have the same length. Chari [2] generalized the definition of the metacommutation permutation to orders in central simple algebras of arbitrary degree and gave results on the cycle structure in the case where the completion of the order is maximal, using results of Fripertinger [7]. In particular, σ ω is interpreted as a permutation on the set of left ideals of reduced norm p.
In this article, we extend results in the quaternion case to allow the completion to be an Eichler order. Let R be a complete discrete valuation ring with field of fractions F and maximal prime ideal p generated by an element p ∈ R, and let O = M 2 (R) ∩ γ −1 M 2 (R)γ be a (local) Eichler R-order in the quaternion algebra B := M 2 (F ), where we define γ = 0 1 p n 0 . We restrict to permuting the set Id(O; p) of principal left ideals of reduced norm p. Each ideal corresponds to a segment in the Bruhat-Tits tree for GL 2 (F ), and we interpret σ ω as an action of O × on the set of segments associated to Id(O; p). We use this action to partition Id(O; p) into two sets, and describe the restriction of σ ω on each set individually.
To describe σ ω further, we define Id(O; p) ′ and Id(M 2 (R); p) ′ to be the set of left ideals of O and M 2 (R), respectively, having reduced norm p, and omitting one specific ideal to be defined later.
there is a partition Id(O; p) ′ = S 1 ⊔ S 2 such that σ ω | S i permutes S i , and there are bijections ϕ : S 1 → Id(M 2 (R); p) ′ and φ γ : S 2 → S 1 such that the following diagrams commute. a.
In this way, we may understand metacommutation in an Eichler order O by studying the given permutations in the two maximal orders containing O. More precisely, up to isomorphism, O ⊆ M 2 (R), and it suffices to understand the permutation given by metacommutation in M 2 (R) itself. Our methods allow us to interpret the metacommutation problem in terms of an action on the Bruhat-Tits tree and hence give a combinatorial description of the cycle structure of σ ω .

Metacommutation setup
We now set up the metacommutation problem and state previous results in this section. Let R be a ring whose field of fractions F is a global field. Let B be a quaternion algebra over F and let O ⊆ B be a quaternion R-order. For a prime p ⊆ R, denote by O p := O ⊗ R R p and B p := B ⊗ F F p the completions of O and B, respectively, at p. Define Id(O; p) to be the set of locally principal left ideals of O having reduced norm p.
Now, it suffices to study the corresponding permutation in the completion O p . The map P → P p := P ⊗ R R p is a bijection between Id(O; p) and the ideals of reduced norm pR p in O p by the local-global dictionary for lattices and [9, Theorem 5.2(iii)]. Using this correspondence and given ω ∈ O, we have We then obtain the same permutation of ideals in O and in O p , so we may focus our attention on the principal left ideals of O p of reduced norm pR p .
Furthermore, we will restrict to permuting only those left ideals of O p that are principal. If a left O p -ideal O p α is principal, then σ ω (O p α) = O p αω + O p p = O p (αω) is also principal since Oαω ⊆ O p p. Therefore, we will still obtain a permutation by restricting the permutation to principal left ideals.
, there is an element a ∈ R × (p) such that aω ∈ O, and P p (aω) = aP p ω = P p ω. We may therefore define σ ω for any ω ∈ O × (p) . In the case where O p is maximal, all left ideals of reduced norm pO p are principal as shown in Reiner [9,Theorem 17.3(iii)]. If O p is in fact maximal, results on the cycle structure of the metacommutation permutation are given by the following two theorems.
Theorem 2.5 (Forsyth-Gurev-Shrima; Chari). If O is a quaternion algebra, then all cycles of σ ω that are not fixed points are the same length; i.e., there is an integer ℓ > 0 such that every cycle either has length either 1 or ℓ.
To set up the metacommutation problem in the Eichler case, we proceed locally. Let R be a complete discrete valuation ring with maximal ideal p = (p), residue field F q := R/p, and field of fractions The reverse containment is obvious.
Recall that for an element ω ∈ O × , we define the permutation σ ω by Given ω ∈ O × , we will describe the cycle structure of σ ω in terms of permutations of the left ideals of O 1 and O 2 of reduced norm p, separately. When rad O ∈ Id(O; p), we show that it is fixed under σ ω for all ω ∈ (R p ) × , and instead restrict σ ω to the set To accomplish this, in the following section we introduce the Bruhat-Tits tree T p for GL 2 (F ).

Ideals of reduced norm p
Eichler [5] and Brzezinski [1] show that the cardinality # Id(O; p) is either 2q or 2q + 1, depending on the level of the order. In this section, we give a combinatorial description of these ideals, together with their explicit generators. We retain notation from the end of Section 2, and start with some known facts about the Bruhat-Tits tree for GL 2 (F ). For more details about the tree, see Serre [11]. Let V := F 2 . We say that two (full) R-lattices L 1 and L 2 in V are homothetic if L 1 = L 2 a for some a ∈ F × . Homothety of lattices is an equivalence relation, which we will denote by [L] = [La]. We can identify B with the ring of linear transformations End F (V ) acting on V on the right, so B × acts transitively on the lattices in V and preserves homothety: We now define the Bruhat-Tits tree T p . The set vertices of T p is given by the homothety classes of latices [L] in V . There is an edge between two vertices if there are lattices L 1 and L 2 in their respective homothety classes such that pL 1 L 2 L 1 . It is known that T p is a (q + 1)-regular tree.
We can interpret the distance between two vertices via their corresponding homothety classes. In particular, given two lattices L 1 and L 2 , by the invariant factor theorem there exists an R-basis which is well-defined on homothety classes. Moreover, the action of B × extends to an action on the tree which preserves δ. In Figure 1, we see a piece of T 2 , the tree for GL 2 (Q 2 ). We turn to the connection between the tree T p and Eichler orders in B. We can identify the maximal order M 2 (R) with End R (L 0 ), which acts on the right on the free R-lattice L 0 := Re 1 + Re 2 , where {e 1 , e 2 } is the standard basis for V . By Reiner [9, (17.3)], each maximal order in B is conjugate to M 2 (R) by an element ξ ∈ B × , and we can identify if and only if End R (L 1 ) = End R (L 2 ), so we have bijections between the set of homothety classes of full R-lattices in V , the set of maximal orders in B, and the vertices in T p . Moreover, the action of B × on T p determines an action of B × on the set of maximal orders corresponding to conjugation.
From now on, we consider non-maximal Eichler orders. Suppose we have an Eichler Then one can associate to O the segment in T p whose endpoints correspond to O 1 and O 2 , and whose length is precisely n. Following from the work of Hijikata [8], O is also the intersection of all the maximal orders contained in this segment. Since we can take O up to conjugation, Then each vertex corresponding to [L 0 γ i ] lies on the segment associated to O (see Figure 2). Denoting γ := γ n , we have We will make frequent use of the following result: Consider a maximal order Λ ⊆ B, corresponding to the vertex X in T p . Then µ ∈ Λ × acts on T p by fixing X.
Proof. Suppose X is given by the homothety class [L].
Since conjugation by B × gives an action on the tree, any order conjugate to O will correspond to a segment of length n as well. We can take the connection between algebraic properties of O and the structure of the tree further, and we have the following connection between the set Id(O; p) and certain segments in the tree.    To prove the backward direction of the correspondence, we use results of Eichler [5] and Brzezinski [1], who proved that the number of principal left ideals of reduced norm p is 2q + 1 when n = 1, and 2q when n ≥ 2. Therefore, it suffices that to show these are also the number of segments in T p satisfying our conditions. Consider the subtree in Figure 3, where O corresponds to the segment XY . There are at most three ways we can obtain a segment of length n with vertices Z and T at distance 1 of X and Y . We could either move XY to the left, to the right, and if n = 1, we could also invert it since δ(X, Y ) = δ(Y, X) = n.
Proposition 3.4 gives rise to the following corollaries, which will help us understand the action of the metacommutation permutation.

Results
In this section, we state our main results on the cycle structure of the permutation given by metacommutation. By Corollary 3.6, when n = 1 the ideal rad(O) is fixed by any σ ω , so it suffices to consider only 2q ideals in Id(O; p). Define Moreover, for each s ∈ R/p, choose b s ∈ R to be any representative of the coset s + p ∈ R/p, and define α s := 1 b s 0 p . Consider the sets of ideals  We shift our attention to S 2 . Since nm(γ −1 α s γ) = p and γ −1 α s γ ∈ O × , S 2 indeed consists of ideals of norm p. Consider the ideal Oγ −1 α s γ, to which we associate the segment with endpoints [L 0 γ −1 α s γ] and [L 0 α s γ]. We claim that this segment is obtained by shifting XY to the right; in particular, we claim that δ([L 0 ], [L 0 α s γ]) = n + 1. Note , and by the definition of δ, we have Finally, we show that each element of the form γ −1 α s γ generates a distinct ideal. Note that Oγ −1 α s γ = Oγ −1 α r γ if and only if γ −1 α s α −1 r γ ∈ O × . On the other hand, γOγ −1 = O, so it follows that α s α −1 r ∈ O × , and a simple calculation shows this happens only if α s = α r .

Example 4.2. Consider the Eichler order
where γ = 0 1 3 0 . Then O corresponds to the segment between the vertices given by [L 0 ] and [L 0 γ] bolded in Figure 5. Corollary 4.1 gives We obtain the segment associated to O where the equalities are coming from the fact that elementary row operations are units in M 2 (R) = End R (L 0 ). The segment associated to Oγ −1 α 1 γ is the dashed segment in Figure 5. . We may suspect that σ ω will fix each of the two sets of ideals determined by S 1 and S 2 . We confirm this in the following corollary. The statement for S 2 follows analogously, this time associating to the ideal Oγ −1 α s γ the segment obtained by shifting XY to the right.
This means that we may view σ ω ∈ sym(S 1 ) × sym(S 2 ). Consider the set of ideals of M 2 (R) given by Define the permutation τ ω by τ ω (P ) = P ω. Note that M 2 (R)γ 1 corresponds to the vertex [L 0 γ 1 ], which is to the immediate right of X.
The ideals in S 1 exactly correspond to those in Id(M 2 (R); p) ′ via the bijection ϕ : Theorem 4.5. We may understand the permutation σ ω by computing σ ω | S 1 and σ ω | S 2 separately, via the following diagrams, which commute. a.
The diagram in (c) is a composition of the first diagram and that in Theorem 4.5, so commutativity follows.
This means that locally, we may write the cycle structure of σ ω in an Eichler order in terms of the cycle structure of two separate permutations given by metacommutation in M 2 (R).
We may then define the following maps: Then, Theorem 4.5 is summarized by the following commutative diagram, which gives a complete description of σ ω in terms of the corresponding cycle structures in the maximal order M 2 (R), which is known (see Forsyth-Gurev-Shrima [6] or Chari [2]).
Corollary 4.6. There is an embedding such that σ(ω) = ι • (τ × τ (γ) )(ω); i.e., the following diagram commutes: In other words, Via the correspondence of S 1 and S 2 in Remark 4.3 and Theorem 4.5(c), one may verify that we get In other words, σ ω permutes the ideals in S 1 by a 3-cycle, and fixes each ideal in S 2 .

Conclusion
We conclude with a more detailed description of the cycle structure of σ ω . Again, suppose that O is an Eichler order of level p n with ω ∈ O × . We now have a description of the cycle structures of the permutation σ ω on S 1 and S 2 , and hence on Id(O; p) in terms of the corresponding cycle structures of the permutations τ ω and τ γ −1 ωγ of Id(M 2 (R); p) ′ . We conclude with a brief discussion of the image and kernel of the map σ in Corollary 4.6. We begin with a lemma about the relationship between ω and γ −1 ωγ.
where c is the coefficient of p n in the bottom left entry of ω.
Proof. If ω ≡ a (mod p), then we may write ω = a + k 1 p bp cp n a + k 2 p .
A quick computation shows that γ −1 ωγ = a + k 1 p c bp n+1 a + k 2 p ≡ a c 0 a (mod p).
Now, by Forsyth-Gurev-Shrima [6], all cycles of τ ω are the same size, and the length is the multiplicative order of ω + p in PGL 2 (R/p). Since σ ω | S 1 has the same cycle structure as τ ω and σ ω | S 2 has the same cycle structure of τ γ −1 ωγ , it is of interest to compare the cycle structure of τ ω and τ γ −1 ωγ . In other words, we seek to compare the multiplicative orders of ω + p and γ −1 ωγ + p in PGL 2 (R/p).
Lemma 5.2. The permutations τ ω and τ γ −1 ωγ have the same number of fixed points, unless one of ω and γ −1 ωγ is a scalar matrix (mod p). Moreover, they each have at most 1 fixed point when neither is a scalar (mod p).
Proof. By modifying the equation for the number of fixed points given by Cohn-Kumar [4] and Chari [2] to exclude the element M 2 (R)γ 1 (which must also be fixed under τ ω following Corollary 3.6), if ω is not a scalar matrix (mod p), the number of fixed points of τ ω is given by the Legendre symbol trd(ω) 2 −4 nrd(ω) p . Writing ω = a b p n c d , we have (a + d) 2 − 4ad − bp n c ≡ (a − d) 2 (mod p), so trd(ω) 2 −4 nrd(ω) p = 0 or 1, depending on whether a ≡ d (mod p) or not. But, since the trace and norm maps are invariant under conjugation (say, by γ n ), the expression is the same for ω and γ −1 ωγ. Remark 5.3. See Example 4.7 for a counterexample in the case where γ −1 ωγ is a scalar matrix (mod p).
We now discuss the kernel of the map σ in Corollary 4.6.
Proof. First, τ ω is the identity permutation if and only if ω ≡ α (mod p) is a scalar (mod p). Therefore, σ ω is the identity permutation if and only if τ ω and τ γ −1 ωγ are both the identity permutation, if and only if ω and γ −1 ωγ are both scalar matrices (mod p). Now, by Lemma 5.1, this holds if and only if ω = a + k 1 p bp cp n+1 a + k 2 p for a, b, c, k 1 , k 2 ∈ R.
Finally, we discuss the image of the map σ in Corollary 4.6.