arXiv subjects

Claudio Bravo

Publications and source records attributed to Claudio Bravo.

15 recordsLinked to original sources

arXiv2026

On some Diophantine and Ergodic properties of the Schneider map over function fields

We introduce and study continued fractions defined by Schneider-like maps over polynomial rings, where the maps are associated with a fixed polynomial of arbitrary degree. In particular, we prove the existence and uniqueness of the continued fraction expansion for every element of the field of Laurent series. We then establish precise Diophantine approximation properties of the corresponding convergents. We also study the dynamical aspects of the underlying map, proving that the Haar measure is invariant and ergodic. As an arithmetic consequence, we determine the asymptotic sum of the digits of the expansion for almost every element. Finally, we identify the set of elements that are worst approximable in this framework and compute its Hausdorff dimension, showing that it is a fractal set of positive dimension.

math.NT↗

arXiv2026

Cohomology of special unitary groups and congruence subgroups

We prove a homotopy invariance result for the first cohomology group of the special unitary group $\mathrm{SU}_3(F[t])$ with coefficients in irreducible representations of $\mathrm{PGL}_2(F)$. The main theorem establishes that this cohomology is naturally isomorphic to the corresponding cohomology of $\mathrm{PGL}_2(F)$.

math.KT↗

arXiv2025

Computing quaternionic representations via twisted forms of Bruhat-Tits trees

This work is devoted to the study of representations of finite subgroups of the group of units of quaternion division algebras over a global or local field arising from the inclusion via extension of scalars splitting the algebra. Following a question by Serre, we study the set $\mathrm{IF}$ of conjugacy classes of integral representations that are conjugates of the given representation over the field. The set $\mathrm{IF}$ is often called the set of integral forms in the literature. In previous works we have seen that, for a given representation, the set $\mathrm{IF}$ can be indexed by the vertex set of a suitable subgraph of the Bruhat-Tits tree for the special linear group. In this work, we describe a construction that allows the simultaneous study of the set $\mathrm{IF}$ over different splitting fields. For this, we devise and use a theory of twisted Galois form of Bruhat-Tits trees. With this tool, we explicitly compute, in most cases, the cardinality of $\mathrm{IF}$ for the representation of the classical quaternion group of order $8$ studied by Serre, Feit and others, as much as for other similar groups.

math.RT↗

arXiv2025

On the homology of special unitary groups over polynomial rings

In this work, we answer the homotopy invariance question for the ''smallest'' non-isotrivial group-scheme over $\mathbb{P}^1$, obtaining a result, which is not contained in previous works due to Knudson and Wendt. More explicitly, let $\mathcal{G}=\mathrm{SU}_{3,\mathbb{P}^1}$ be the (non-isotrivial) non-split group-scheme over $\mathbb{P}^1$ defined from the standard (isotropic) hermitian form in three variables. In this article, we prove that there exists a natural homomorphism $\mathrm{PGL}_2(F) \to \mathcal{G}(F[t])$ that induces isomorphisms $H_*(\mathrm{PGL}_2(F), \mathbb{Z}) \to H_*(\mathcal{G}(F[t]), \mathbb{Z})$. Then we study the rational homology of $\mathcal{G}(F[t,t^{-1}])$, by previously describing suitable fundamental domains for certain arithmetic subgroups of $\mathcal{G}$.

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arXiv2024

Tessellations of an affine apartment by affine weight polytopes

Let $\A$ be a finite dimensional vector space and $\Phi$ be a finite root system in $\A$. To this data is associated an affine poly-simplicial complex. Motivated by a forthcoming construction of connectified higher buildings, we study "affine weight polytopes" associated to these data. We prove that these polytopes tesselate $\A$. We also prove a kind of "mixed" tessellation, involving the affine weight polytopes and the poly-simplical structure on $\A$.

math.GR↗

arXiv2024

A Continued Fractions Theory for the completion of the Puiseux field

In this work, we study a continued fractions theory for the topological completion of the field of Puiseux series. As usual, we prove that any element in the completion can be developed as a unique continued fractions, whose coefficients are polynomials in roots of the variable, and that this approximation is the best ''rational'' Diophantine approximation of such element. Then, we interpret the preceding result in terms of the action of a suitable arithmetic subgroup of the special linear group on the Berkovich space defined over the said completion. We also explore the connections between points of type IV of the Berkovich space in terms of some ''non-convergent'' or ''undefined'' continued fractions, in a sense that we make precise in the text.

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arXiv2024

Diophantine Approximation in local function fields via Bruhat-Tit trees

We use the theory of arithmetic quotients of the Bruhat-Tits tree developed by Serre and others to obtain Dirichlet-style theorems for Diophantine approximation on global function fields. This approach allows us to find sharp values for the constants involved and, occasionally, explicit examples of badly approximable quadratic irrationals. Additionally, we can use this method to easily compute the measure of the set of elements that can be written as the limit of a sequence of ``better than expected'' approximants. All these results can be easily obtained via continued fractions when they are available, so that quotient graphs can be seen as a partial replacement of them when this fails to be the case.

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arXiv2023

Arithmetic subgroups of Chevalley group schemes over function fields II: Conjugacy classes of maximal unipotent subgroups

Let $\mathcal{C}$ be a smooth, projective, geometrically integral curve defined over a perfect field $\mathbb{F}$. Let $k=\mathbb{F}(\mathcal{C})$ be the function field of $\mathcal{C}$. Let $\mathbf{G}$ be a split simply connected semisimple $\mathbb{Z}$-group scheme. Let $\mathcal{S}$ be a finite set of places of $\mathcal{C}$. In this paper, we investigate on the conjugacy classes of maximal unipotent subgroups of $\mathcal{S}$-arithmetic subgroups. These are parameterized thanks to the Picard group of $\mathcal{O}_{\mathcal{S}}$ and the rank of $\mathbf{G}$. Furthermore, these maximal unipotent subgroups can be realized as the unipotent part of natural stabilizer, which are the stabilizers of sectors of the associated Bruhat-Tits building. We decompose these natural stabilizers in terms of their diagonalisable part and unipotent part, and we precise the group structure of the diagonalisable part.

math.GR↗

arXiv2023

Arithmetic subgroups of Chevalley group schemes over function fields I: quotients of the Bruhat-Tits building by $\{P\}$-arithmetic subgroups

Let $\mathbf{G}$ be a reductive Chevalley group scheme (defined over $\mathbb{Z}$). Let $\mathcal{C}$ be a smooth, projective, geometrically integral curve over a field $\mathbb{F}$. Let $P$ be a closed point on $\mathcal{C}$. Let $A$ be the ring of functions that are regular outside $\lbrace P \rbrace$. The fraction field $k$ of $A$ has a discrete valuation $ν=ν_{P}: k^{\times} \rightarrow \mathbb{Z}$ associated to $P$. In this work, we study the action of the group $ \textbf{G}(A)$ of $A$-points of $\mathbf{G}$ on the Bruhat-Tits building $\mathcal{X}=\mathcal{X}(\textbf{G},k,ν_{P})$ in order to describe the structure of the orbit space $ \textbf{G}(A)\backslash \mathcal{X}$. We obtain that this orbit space is the ``gluing'' of a closed connected CW-complex with some sector chambers. The latter are parametrized by a set depending on the Picard group of $\mathcal{C} \smallsetminus \{P\}$ and on the rank of $\mathbf{G}$. Moreover, we observe that any rational sector face whose tip is a special vertex contains a subsector face that embeds into this orbit space.

math.GR↗

arXiv2023

Relative homology of arithmetic subgroups of $\mathrm{SU}(3)$

Let $\mathcal{C}$ be a smooth, projective and geometrically integral curve defined over a finite field $\mathbb{F}$. Let $A$ be the ring of function of $\mathcal{C}$ that are regular outside a closed point $P$ and let $k=\mathrm{Quot}(A)$. Let $\mathcal{G}=\mathrm{SU}(3)$ be the non-split group-scheme defined from an (isotropic) hermitian form in three variables. In this work, we describe, in terms of the Euler-Poincaré characteristic, the relative homology groups of certain arithmetic subgroups $G$ of $\mathcal{G}(A)$ modulo a representative system $\mathfrak{U}$ of the conjugacy classes of their maximal unipotent subgroups. In other words, we measure how far are the homology groups of $G$ from being the coproducts of the corresponding homology groups of the subgroups $U \in \mathfrak{U}$.

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arXiv2022

Quotients of the Bruhat-Tits tree by function field analogs of the Hecke congruence subgroups

Let C be a smooth, projective and geometrically integral curve defined over a finite field F. For each closed point P of C, let R be the ring of functions that are regular outside P, and let K be the completion at P of the function field of C. In order to study groups of the form GL2(R), Serre describes the quotient graph GL2(R)\t, where t is the Bruhat-Tits tree defined from SL2(K). In particular, Serre shows that GL2(R)\t is the union of a finite graph and a finite number of ray shaped subgraphs, which are called cusps. It is not hard to see that finite index subgroups inherit this property. In this work we describe the associated quotient graph H\t for the action on t of the group H of matrices in GL2(R) that are upper triangular modulo a certain ideal I of R. More specifically, we give a explicit formula for the cusp number of H\t. Then, by using Bass-Serre Theory, we describe the combinatorial structure of H. These groups play, in the function field context, the same role as the Hecke congruence subgroups of SL2(Z).

math.GR↗

arXiv2021

Quotients of the Bruhat-Tits tree by arithmetic subgroups of special unitary groups

Let $K$ be the function field of a curve $C$ over a field $\mathbb{F}$ of either odd or zero characteristic. Following the work by Serre and Mason on $\mathrm{SL}_2$, we study the action of arithmetic subgroups of $\mathrm{SU}(3)$ on its corresponding Bruhat-Tits tree associated to a suitable completion of $K$. More precisely, we prove that the quotient graph "looks like a spider", in the sense that it is the union of a set of cuspidal rays (the "legs"), parametrized by an explicit Picard group, that are attached to a connected graph (the "body"). We use this description in order to describe these arithmetic subgroups as amalgamated products and study their homology. In the case where $\mathbb{F}$ is a finite field, we use a result by Bux, Köhl and Witzel in order to prove that the "body" is a finite graph, which allows us to get even more precise applications.

math.GR↗

arXiv2020

Branches in the Bruhat-Tits tree for local fields of even characteristic

We extend our previous computations for the relative positions of branches of quaternions to the case of local fields of even characteristic. This is a key step to understand the set of maximal orders containing a given suborder, which is useful, for instance, to compute relative spinor images, thus solving the selectivity problem. In our previous work, the results where given in terms of the quadratic defect. In the present context, we introduce and characterize an analogous concept for Artin-Schreier extensions. It is no longer useful to restrict our attention to orders generated by pure quaternions, as a separable quadratic extension contains no non-trivial element of null trace. In this work we state our result for an arbitrary pair of generators, for which we discuss a more general version of the Hilbert symbol in this context.

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arXiv2019

On the missing branches of the Bruhat-Tits tree

Let k be a local field and let A be the two-by-two matrix algebra over k. In our previous work we developed a theory that allows the computation of the set of maximal orders in A containing a given suborder. This set is given as a sub-tree of the Bruhat-Tits tree that is called the branch of the order. Branches have been used to study the global selectivity problem and also to compute local embedding numbers. They can usually be described in terms of two invariants. To compute these invariants explicitly, the strategy in our past work has been visualizing branches through the explicit representation of the Bruhat-Tits tree in terms of balls in k. This is easier for orders spanning a split commutative sub-algebra, i.e., an algebra isomorphic to (k x k). In the present work, we develop a theory of branches over field extension that can be used to extend our previous computations to orders spanning a field. We use the same idea to compute branches for orders generated by arbitrary pairs of non-nilpotent pure quaternions. In fact, the hypotheses on the generators are not essential.

math.NT↗

arXiv2019

On genera containing non-split Eichler orders over function fields

Grothendieck-Birkhoff Theorem states that every finite dimensional vector bundle over the projective line P1 splits as the sum of one dimensional vector bundles. This can be rephrased, in terms of orders, as stating that all maximal orders over the projective line in a matrix algebra split. In this work we study the extent to which this result can be generalized to Eichler orders when the base field F is finite. To be precise, we characterize both the genera of Eichler orders containing only split orders and the genera containing only a finite number of non-split conjugacy classes. The latter characterization is given for arbitrary projective curves over F. The method developed here also allows us to compute quotient graphs for some subgroups of $PGL_2(F[t])$ of arithmetical interest. This paper includes material from the unpublished work "Simultaneous diagonalization of vector bundles".

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