Bruhat-Tits Tree Research Articles

Computing fundamental domains for the Bruhat–Tits tree for , -adic automorphic forms, and the canonical embedding of Shimura curves

AbstractWe describe algorithms that allow the computation of fundamental domains in the Bruhat–Tits tree for the action of discrete groups arising from quaternion algebras. These algorithms are used to compute spaces of rigid modular forms of arbitrary even weight, and we explain how to evaluate such forms to high precision using overconvergent methods. Finally, these algorithms are applied to the calculation of conjectural equations for the canonical embedding of p-adically uniformizable rational Shimura curves. We conclude with an example in the case of a genus 4 Shimura curve.

THE STABILIZERS IN A DRINFELD MODULAR GROUP OF THE VERTICES OF ITS BRUHAT–TITS TREE: AN ELEMENTARY APPROACH

Let K be an algebraic function field of one variable with constant field k and let [Formula: see text] be the Dedekind domain consisting of all those elements of K which are integral outside a fixed place ∞ of K. When k is finite the group [Formula: see text] plays a central role in the theory of Drinfeld modular curves analogous to that played by SL2(ℤ) in the classical theory of modular forms. When k is finite (respectively, infinite) we refer to a group [Formula: see text] as an arithmetic (respectively, non-arithmetic) Drinfeld modular group. Associated with [Formula: see text] is its Bruhat–Tits tree, [Formula: see text]. The structure of the group is derived from that of the quotient graph [Formula: see text]. Using an elementary approach which refers explicitly to matrices we determine the structure of all the vertex stabilizers of [Formula: see text]. This extends results of Serre, Takahashi and the authors. We also determine all possible valencies of the vertices of [Formula: see text] for the important special case where ∞ has degree 1.

Graphs of Hecke operators

Let $X$ be a curve over $\F_q$ with function field $F$. In this paper, we define a graph for each Hecke operator with fixed ramification. A priori, these graphs can be seen as a convenient language to organize formulas for the action of Hecke operators on automorphic forms. However, they will prove to be a powerful tool for explicit calculations and proofs of finite dimensionality results. We develop a structure theory for certain graphs $G_x$ of unramified Hecke operators, which is of a similar vein to Serre's theory of quotients of Bruhat Tits trees. To be precise, $G_x$ is locally a quotient of a Bruhat Tits tree and has finitely many components. An interpretation of $G_x$ in terms of rank 2 bundles on $X$ and methods from reduction theory show that $G_x$ is the union of finitely many cusps, which are infinite subgraphs of a simple nature, and a nucleus, which is a finite subgraph that depends heavily on the arithmetics of $F$. We describe how one recovers unramified automorphic forms as functions on the graphs $G_x$. In the exemplary cases of the cuspidal and the toroidal condition, we show how a linear condition on functions on $G_x$ leads to a finite dimensionality result. In particular, we re-obtain the finite-dimensionality of the space of unramified cusp forms and the space of unramified toroidal automorphic forms. In an Appendix, we calculate a variety of examples of graphs over rational function fields.

On computing quaternion quotient graphs for function fields

Let Λ be a Fq[T ]-maximal order in a division quaternion algebra over Fq(T ) which is split at the place∞. The present article gives an algorithm to compute a fundamental domain for the action of the group of units Λ∗ on the Bruhat-Tits tree T associated to PGL2(Fq((1/T ))). This action is a function field analog of the action of a co-compact Fuchsian group on the upper half plane. The algorithm also yields an explicit presentation of the group Λ∗ in terms of generators and relations. Moreover we determine an upper bound for its running time using that Λ∗\T is almost Ramanujan.

P-torsion coefficient systems for [formula omitted] and [formula omitted

We show that the categories of smooth SL 2 ( Q p ) -representations (resp. GL 2 ( Q p ) -representations) of level 1 on p-torsion modules are equivalent with certain explicitly described equivariant coefficient systems on the Bruhat–Tits tree; the coefficient system assigned to a representation V assigns to an edge τ the invariants in V under the pro- p-Iwahori subgroup corresponding to τ. The proof relies on computations of the group cohomology of a compact open subgroup group N 0 of the unipotent radical of a Borel subgroup.

Explicit description of irreducible [formula omitted]-representations over [formula omitted

Let p be an odd prime number. The classification of irreducible representations of GL 2 ( Q p ) over F ¯ p is known thanks to the works of Barthel and Livné (1995) [BL95] and Breuil (2003) [Bre03a]. In the present paper we illustrate an exhaustive description of such irreducible representations, through the study of certain functions on the Bruhat–Tits tree of GL 2 ( Q p ) . In particular, we are able to detect the socle filtration for the KZ-restriction of supersingular representations, principal series and special series.

Ramanujan bigraphs associated with $SU(3)$ over a $p$-adic field

We use the representation theory of the quasisplit form G of SU(3) over a p-adic field to investigate whether certain quotients of the Bruhat-Tits tree associated to this form are Ramanujan bigraphs. We show that a quotient of the tree associated with G (which is a biregular bigraph) is Ramanujan if and only if G satisfies a Ramanujan type conjecture. This result is analogous to the seminal case of PGL 2 (Q p ) considered by Lubotzky, Phillips, and Sarnak. As a consequence, the classification of the automorphic spectrum of the unitary group in three variables by Rogawski implies the existence of certain infinite families of Ramanujan bigraphs.

Fundamental domains for congruence subgroups of [formula omitted] in positive characteristic

In this work, we construct fundamental domains for congruence subgroups of SL 2 ( F q [ t ] ) and PGL 2 ( F q [ t ] ) . Our method uses Gekeler's description of the fundamental domains on the Bruhat–Tits tree X = X q + 1 in terms of cosets of subgroups. We compute the fundamental domains for a number of congruence subgroups explicitly as graphs of groups using the computer algebra system Magma.

Nonrational Genus Zero Function Fields and the Bruhat–Tits Tree

Let K be a function field with constant field k, and let ∞ be a fixed place of K. Let 𝒞 be the Dedekind domain consisting of all those elements of K which are integral outside ∞. The group G = GL 2(𝒞) is important for a number of reasons. For example, when k is finite, it plays a central role in the theory of Drinfeld modular curves. Many properties follow from the action of G on its associated Bruhat–Tits tree, 𝒯. Classical Bass–Serre theory shows how a presentation for G can be derived from the structure of the quotient graph (or fundamental domain) G \\ 𝒯. The shape of this quotient graph (for any G) is described in a fundamental result of Serre. However, there are very few known examples for which a detailed description of G \\ 𝒯 is known. (One such is the rational case, 𝒞 =k[t], i.e., when K has genus zero, and ∞ has degree one.) In this article, we give a precise description of G \\ 𝒯 for the case where the genus of K is zero, K has no places of degree one, and ∞ has degree two. Among the known examples a new feature here is the appearance of vertex stabilizer subgroups (of G) which are of quaternionic type.

Mumford dendrograms and discrete p-adic symmetries

In this article, we present an effective encoding of dendrograms by embedding them into the Bruhat-Tits trees associated to p-adic number fields. As an application, we show how strings over a finite alphabet can be encoded in cyclotomic extensions of ℚ p and discuss p-adic DNA encoding. The application leads to fast p-adic agglomerative hierarchic algorithms similar to the ones recently used e.g. by A. Khrennikov and others. From the viewpoint of p-adic geometry, to encode a dendrogram X in a p-adic field K means to fix a set S of K-rational punctures on the p-adic projective line ℙ1. To ℙ1 \ S is associated in a natural way a subtree inside the Bruhat-Tits tree which recovers X, a method first used by F. Kato in 1999 in the classification of discrete subgroups of PGL2(K). Next, we show how the p-adic moduli space \( \mathfrak{M}_{0,n} \) of ℙ1 with n punctures can be applied to the study of time series of dendrograms and those symmetries arising from hyperbolic actions on ℙ1. In this way, we can associate to certain classes of dynamical systems a Mumford curve, i.e. a p-adic algebraic curve with totally degenerate reduction modulo p. Finally, we indicate some of our results in the study of general discrete actions on ℙ1, and their relation to p-adic Hurwitz spaces.

On non-uniform Ramanujan complexes

where Ax is the adjacency matrix of the graph X. Such graphs were constructed there as quotients of the Bruhat-Tits tree associated to G = PGL2(QP), modulo congruence subgroups of uniform (i.e. cocompact) lattices of G, where Qp is the field of the p-adic integers. More examples were given by Morgenstern [Ml] by replacing Qp by a local field F of characteristic p > 0, such as ¥q((t)). One significant difference is that unlike PGL2(QP), PGL»2(Fg ((£))) has a nonuniform lattice A = PGIj2(Fg[l/t]), which is of finite covolume out not cocompact. For congruence subgroups of A, the quotient graphs are infinite, but the edges and vertices come with weights, so that the total weight is finite. Morgenstern [M2] went ahead to define Ramanujan diagrams as such weighted objects satisfying Equation (1.1), where this time Ax is the 'weighted adjacency matrix' (see Section 2 for more details) . He constructed explicit examples and even showed a surprising application of them to the theory of networks [M3] . Recently, the concept of Ramanujan graphs was generalized to higher dimensional complexes [Li, LSV1, Sa] (see Remark 4.5.(2)) by replacing PGL/2(F) by PGLd(F) for d > 2. This time one requires bounds on the simultaneous spectrum of d 1 'Hecke operators', which are 'colored adjacency matrices'; see Section 3 below.

Degenerating Families of Dendrograms

Dendrograms used in data analysis are ultrametric spaces, hence objects of nonarchimedean geometry. It is known that there exist p-adic representations of dendrograms. Completed by a point at infinity, they can be viewed as subtrees of the Bruhat-Tits tree associated to the p-adic projective line. The implications are that certain moduli spaces known in algebraic geometry are in fact p-adic parameter spaces of dendrograms, and stochastic classification can also be handled within this framework. At the end, we calculate the topology of the hidden part of a dendrogram.

Mumford Dendrograms

An effective $p$-adic encoding of dendrograms is presented through an explicit embedding into the Bruhat-Tits tree for a $p$-adic number field. This field depends on the number of children of a vertex and is a finite extension of the field of $p$-adic numbers. It is shown that fixing $p$-adic representatives of the residue field allows a natural way of encoding strings by identifying a given alphabet with such representatives. A simple $p$-adic hierarchic classification algorithm is derived for $p$-adic numbers, and is applied to strings over finite alphabets. Examples of DNA coding are presented and discussed. Finally, new geometric and combinatorial invariants of time series of $p$-adic dendrograms are developped.

The congruence kernel of an arithmetic lattice in a rank one algebraic group over a local field

Let k be a global field and let kυ be the completion of k with respect to υ a non-archimedean place of k. Let G be a connected, simply-connected algebraic group over k, which is absolutely almost simple of kυ-rank 1. Let G = G(kυ). Let Г be an arithmetic lattice in G and let C = C(Г) be its congruence kernel. Lubotzky has shown that C is infinite, confirming an earlier conjecture of Serre. Here we provide complete solution of the congruence subgroup problem for Г by determining the structure of C. It is shown that C is a free profinite product, one of whose factors is , the free profinite group on countably many generators. The most surprising conclusion from our results is that the structure of C depends only on the characteristic of k. The structure of C is already known for a number of special cases. Perhaps the most important of these is the (non-uniform) example , where is the ring of S-integers in k, with S = {υ}, which plays a central role in the theory of Drinfeld modules. The proof makes use of a decomposition theorem of Lubotzky, arising from the action of Г on the Bruhat-Tits tree associated with G.

Dynamique sur le rayon modulaire et fractions continues en caractéristique p

Let K̂ be the field of formal Laurent series in X−1 over the finite field k, and let A be the ring of polynomials in X over k. One of the main results of the paper is to give a natural coding of the (discrete) geodesic flow on the quotient of the Bruhat–Tits tree 𝕋 of PGL2(K̂) by PGL2(A), using the continued fraction expansion of the endpoints of the geodesic lines in 𝕋 (the space of ends of 𝕋 identifies with ℙ 1(K̂). In particular, the invariance of the Haar measure by the Artin transformation can be deduced from the invariance of the Bowen–Margulis measure by the geodesic flow.

Cusps of Lattices in Rank 1 Lie Groups over Local Fields

Let G be the group of rational points of a semisimple algebraic group of rank 1 over a non-Archimedean local field. We improve upon Lubotzky's analysis of graphs of groups describing the action of lattices in G on its Bruhat–Tits tree assuming a condition on unipotents in G. The condition holds for all but a few types of rank 1 groups. A fairly straightforward simplification of Lubotzky's definition of a cusp of a lattice is the key step to our results. We take the opportunity to reprove Lubotzky's part in the analysis from this foundation.

Un revêtement de l'arbre de GL2 d'un corps local

Let F be a non-archimedean local field with residue class field k. Put G=GL2(F), Γ=PGL2(k) and let X denote the Bruhat–Tits tree of G. We construct a one-dimensional simplicial complex \(\tilde X\), equipped with an action of G × Γ and with a G × Γ-equivariant simplicial projection \(\pi :\tilde X \to X\) (for the trivial action of Γ on X). We prove that the cohomology with compact support \(H_c^1 (\tilde X,\mathbb{C})\) contains nontrivial representations of G (in particular positive level supercuspidal representations).

Discontinuous groups in positive characteristic and automorphisms of Mumford curves

A Mumford curve of genus g (>1) over a non-archimedean valued field k of positive characteristic has at most max{12(g-1), 2 g^(1/2) (g^(1/2)+1)^2} automorphisms. This bound is sharp in the sense that there exist Mumford curves of arbitrary high genus that attain it (they are fibre products of suitable Artin-Schreier curves). The proof provides (via its action on the Bruhat-Tits tree) a classification of discontinuous subgroups of PGL(2,k) that are normalizers of Schottky groups of Mumford curves with more than 12(g-1) automorphisms. As an application, it is shown that all automorphisms of the moduli space of rank-2 Drinfeld modules with principal level structure preserve the cusps.

On Zeta Functions of Arithmetically Defined Graphs

We study the graphX(n) that is defined as the finite part of the quotient Γ(n)\\T, with T the Bruhat–Tits tree over Fq((1/T)) and Γ(n) the principal congruence subgroup of Γ=GL2(Fq[T]) of leveln∈Fq[T]. We give concrete realizations of theL-functions of the finite part of the halfline Γ\\T for finite unitary representations of Γ that factor over Γ(n),nprime. This allows us to give explicit formulae for the zeta function ofX(n) for smalln. As an application, we show that these graphs are very good concentrators. Moreover, we construct a new unbounded family of Ramanujan graphs, considering regularizations ofX(n).

Arithmetically Defined Representations of Groups of TypeSL(2, [formula omitted]q)

We consider a groupGlying between the special and the general linear group of order 2 over the finite field Fq. A representation which arises from a natural action ofGon a quotient graph of the Bruhat–Tits tree ofGL(2, Fq((π))) is described and its irreducible components are determined.