Mumford Curves Research Articles

Bi-algebraicity in the rank one Riemann-Hilbert correspondence via o-minimality

For a smooth, projective, complex algebraic variety X, the Riemann–Hilbert correspondence establishes a complex analytic isomorphism between the ‘Betti moduli space’ of rank n local systems on Xan and the ‘de Rham moduli space’ of rank n vector bundles with flat connection on X. In the rank one case, C. Simpson precisely characterizes the subvarieties of these moduli spaces that are ‘bi-algebraic’ for this typically transcendental, analytic isomorphism. In this short note, we give a new proof of this characterization of Simpson, using methods from o-minimal geometry. We adapt the o-minimal proof to a p-adic setting, namely that of Mumford curves.

Mirror symmetry and Hitchin system on Deligne–Mumford curves: Strominger–Yau–Zaslow duality

Abstract We systematically study the moduli stacks of Higgs bundles, spectral curves, and Norm maps on Deligne–Mumford curves. As an application, under some mild conditions, we prove the Strominger–Yau–Zaslow duality for the moduli spaces of Higgs bundles over a hyperbolic stacky curve.

Heat Equations and Wavelets on Mumford Curves and Their Finite Quotients

A class of heat operators over non-archimedean local fields acting on L_2-function spaces on holed discs in the local field are developed and seen as being operators previously introduced by Zúñiga-Galindo, and if the underlying trees are regular, they are associated here with certain finite Kronecker product graphs. L_2-spaces and integral operators invariant under the action of a finite group acting on a holed disc are studied, and then applied to Mumford curves. It is found that the spectral gap in families of Mumford curves can become arbitrarily small.

A p-adic Maass–Shimura operator on Mumford curves

We study a p-adic Maass–Shimura operator in the context of Mumford curves defined by [15]. We prove that this operator arises from a splitting of the Hodge filtration, thus answering a question in [15]. We also study the relation of this operator with generalized Heegner cycles, in the spirit of [1, 4, 19, 28].

A non-Archimedean analogue of Teichm\xfcller space and its tropicalization

In this article we use techniques from tropical and logarithmic geometry to construct a non-Archimedean analogue of Teichmüller spaceoverline{{{mathcal {T}}}}_g whose points are pairs consisting of a stable projective curve over a non-Archimedean field and a Teichmüller marking of the topological fundamental group of its Berkovich analytification. This construction is closely related to and inspired by the classical construction of a non-Archimedean Schottky space for Mumford curves by Gerritzen and Herrlich. We argue that the skeleton of non-Archimedean Teichmüller space is precisely the tropical Teichmüller space introduced by Chan–Melo–Viviani as a simplicial completion of Culler–Vogtmann Outer space. As a consequence, Outer space turns out to be a strong deformation retract of the locus of smooth Mumford curves in overline{{mathcal {T}}}_g.

Tame torsion and the tame inverse Galois problem

Fix a positive integer g and a squarefree integer m. We prove the existence of a genus g curve C/{mathbb {Q}} such that the mod m representation of its Jacobian is tame. The method is to analyse the period matrices of hyperelliptic Mumford curves, which could be of independent interest. As an application, we study the tame version of the inverse Galois problem for symplectic matrix groups over finite fields.

Constructing smooth and fully faithful tropicalizations for Mumford curves

The tropicalization of an algebraic variety X is a combinatorial shadow of X, which is sensitive to a closed embedding of X into a toric variety. Given a good embedding, the tropicalization can provide a lot of information about X. We construct two types of these good embeddings for Mumford curves: fully faithful tropicalizations, which are embeddings such that the tropicalization admits a continuous section to the associated Berkovich space X^{{{,mathrm{an},}}} of X, and smooth tropicalizations. We also show that a smooth curve that admits a smooth tropicalization is necessarily a Mumford curve. Our key tool is a variant of a lifting theorem for rational functions on metric graphs.

Generalised Diffusion on Moduli Spaces of p-Adic Mumford Curves

A construction of a pseudo-differential operator on non-archimedean local fields invariant under a finite group action is given together with the solution of the corresponding Cauchy problem. This construction is applied to parts of the Gerritzen-Herrlich Teichm\"uller space in order to obtain a self-adjoint operator whose spectrum can decide about certain properties of the reduction graph of the corresponding Mumford curves.

Generalized Heegner cycles on Mumford curves

We study generalised Heegner cycles, originally introduced by Bertolini–Darmon–Prasanna for modular curves in Bertolini et al. (Duke Math J 162(6):1033–1148, 2013), in the context of Mumford curves. The main result of this paper relates generalized Heegner cycles with the two variable anticyclotomic p-adic L-function attached to a Coleman family $$f_\infty $$ and an imaginary quadratic field K, constructed in Bertolini and Darmon (Invent Math 168(2):371–431, 2007) and Seveso (J Reine Angew Math 686:111–148, 2014). While in Bertolini and Darmon (Invent Math 168(2):371–431, 2007) and Seveso (J Reine Angew Math 686:111–148, 2014) only the restriction to the central critical line of this 2 variable p-adic L-function is considered, our generalised Heegner cycles allow us to study the restriction of this function to non-central critical lines. The main result expresses the derivative along the weight variable of this anticyclotomic p-adic L-function restricted to non necessarily central critical lines as a combination of the image of generalized Heegner cycles under a p-adic Abel–Jacobi map. In studying generalised Heegner cycles in the context of Mumford curves, we also obtain an extension of a result of Masdeu (Compos Math 148(4):1003–1032, 2012) for the (one variable) anticyclotomic p-adic L-function of a modular form f and K at non-central critical integers.

Stratified Bundles on Curves and Differential Galois Groups in Positive Characteristic

Stratifications and iterative differential equations are analogues in positive characteristic of complex linear differential equations. There are few explicit examples of stratifications. The main goal of this paper is to construct stratifications on projective or affine curves in positive characteristic and to determine the possibilities for their differential Galois groups. For the related "differential Abhyankar conjecture" we present partial answers, supplementing the literature. The tools for the construction of regular singular stratifications and the study of their differential Galois groups are $p$-adic methods and rigid analytic methods using Mumford curves and Mumford groups. These constructions produce many stratifications and differential Galois groups. In particular, some information on the tame fundamental groups of affine curves is obtained.

Correction to: Discontinuous groups in positive characteristic and automorphisms of Mumford curves

The main theorem of [2], claiming to give an upper bound for the number of automorphisms of a Mumford curve in characteristic

Comparison of two notions of subharmonicity on non-archimedean curves

We show that a continuous function on the analytification of a smooth proper algebraic curve over a non-archimedean field is subharmonic in the sense of Thuillier if and only if it is psh, i.e. subharmonic in the sense of Chambert-Loir and Ducros. This equivalence implies that the property psh for continuous functions is stable under pullback with respect to morphisms of curves. Furthermore, we prove an analogue of the monotone regularization theorem on the analytification of $$\mathbb {P}^{1}$$ and Mumford curves using this equivalence.

Mumford curves and Mumford groups in positive characteristic

A Mumford group is a discontinuous subgroup Γ of PGL2(K), where K denotes a non archimedean valued field, such that the quotient by Γ is a curve of genus 0. As abstract group Γ is an amalgam of a finite tree of finite groups. For K of positive characteristic the large collection of amalgams having two or three branch points is classified. Using these data Mumford curves with a large group of automorphisms are discovered. A long combinatorial proof, involving the classification of the finite simple groups, is needed for establishing an upper bound for the order of the group of automorphisms of a Mumford curve. Orbifolds in the category of rigid spaces are introduced. For the projective line the relations with Mumford groups and singular stratified bundles are studied. This paper is a sequel to [26]. Part of it clarifies, corrects and extends work of G. Cornelissen, F. Kato and K. Kontogeorgis.

Mumford curves covering 𝑝-adic Shimura curves and their fundamental domains

We give an explicit description of fundamental domains associated with the p p -adic uniformisation of families of Shimura curves of discriminant D p Dp and level N ≥ 1 N\geq 1 , for which the one-sided ideal class number h ( D , N ) h(D,N) is 1 1 . The results obtained generalise those in Schottky groups and Mumford curves, Springer, Berlin, 1980 for Shimura curves of discriminant 2 p 2p and level N = 1 N=1 . The method we present here enables us to find Mumford curves covering Shimura curves, together with a free system of generators for the associated Schottky groups, p p -adic good fundamental domains, and their stable reduction-graphs. The method is based on a detailed study of the modular arithmetic of an Eichler order of level N N inside the definite quaternion algebra of discriminant D D , for which we generalise the classical results of Hurwitz. As an application, we prove general formulas for the reduction-graphs with lengths at p p of the families of Shimura curves considered.

Cyclic coverings of the projective line by Mumford curves in positive characteristic

We study the rigid analytic geometry of cyclic coverings of the projective line. We determine the defining equation of cyclic coverings of degree $p$ of the projective line by Mumford curves over complete discrete valuation fields of positive characteristic $p$. Previously, Bradley studied that of any degree over non-archimedean local fields of characteristic zero.

Poincaré duality for the tropical Dolbeault cohomology of non-archimedean Mumford curves

We calculate the tropical Dolbeault cohomology for the analytifications of P1 and Mumford curves over non-archimedean fields. We show that the cohomology satisfies Poincaré duality and behaves analogously to the cohomology of curves over the complex numbers. Further, we give a complete calculation of the dimension of the cohomology on a basis of the topology.

A nonarchimedean Ax–Lindemann theorem

We prove a statement of Ax-Lindemann type for the uniformization of products of Mumford curves whose associated fundamental groups are non-abelian Schottky subgroups of $\mathop{\rm PGL}(2,\bar{\mathbf Q_p})$ contained in $\mathop{\rm PGL}(2,\bar{\mathbf Q})$. In particular, we characterize bi-algebraic irreducible subvarieties of the uniformization.

Green’s functions on Mumford curves

We study an analogue of a Green’s function on Mumford curves. The meromorphic continuation of this Green’s function comes from a “differential equation,” and we interpret the special value at \(s=0\) as a “volume” of the corresponding curve. Using harmonic analysis on Bruhat-Tits trees, we derive a “Kronecker limit formula” for the Green’s functions in question, which connects the derivative at \(s=0\) with the Manin–Drinfeld theta functions. Following Gross’ description of Neron’s local height pairing on Mumford curves, the special derivative here then equals twice of the Neron’s local height with sign changed.

Generalized Artin–Mumford curves over finite fields

Let Fq be the finite field of order q=ph with p>2 prime and h>1, and let Fq¯ be a subfield of Fq. From any two q¯-linearized polynomials L1,L2∈F‾q[T] of degree q, we construct an ordinary curve X(L1,L2) of genus g=(q−1)2 which is a generalized Artin–Schreier cover of the projective line P1. The automorphism group of X(L1,L2) over the algebraic closure F‾q of Fq contains a semidirect product Σ⋊Γ of an elementary abelian p-group Σ of order q2 by a cyclic group Γ of order q¯−1. We show that for L1≠L2, Σ⋊Γ is the full automorphism group Aut(X(L1,L2)) over F‾q; for L1=L2 there exists an extra involution and Aut(X(L1,L1))=Σ⋊Δ with a dihedral group Δ of order 2(q¯−1) containing Γ. Two different choices of the pair {L1,L2} may produce birationally isomorphic curves, even for L1=L2. We prove that any curve of genus (q−1)2 whose F‾q-automorphism group contains an elementary abelian subgroup of order q2 is birationally equivalent to X(L1,L2) for some separable q¯-linearized polynomials L1,L2 of degree q. We produce an analogous characterization in the special case L1=L2. This extends a result on the Artin–Mumford curves, due to Arakelian and Korchmáros [1].

KMS weights on higher rank buildings

We extend some of the results of Carey-Marcolli-Rennie on modular index invariants of Mumford curves to the case of higher rank buildings. We discuss notions of KMS weights on buildings, that generalize the construction of graph weights over graph C*-algebras.