Non-Archimedean Schottky Uniformization

• Non-archimedean Schottky uniformization is the analytic method that constructs Mumford curves through the free action of Schottky groups on the Berkovich projective line.
• It organizes the moduli of these curves by linking explicit Schottky group parameters with tropical geometry and non-archimedean Hodge theory.
• Advanced computational techniques, including non-archimedean theta functions and iterative algorithms, enable efficient determination of curve invariants and group actions.

Non-Archimedean Schottky uniformization is the theory and framework describing the analytic uniformization of certain algebraic curves—so-called Mumford curves—over non-archimedean valued fields by the action of discrete, free, finitely generated, torsion-free subgroups of $\mathrm{PGL}_2(k)$ (Schottky groups). The construction encodes both the analytic and arithmetic structure of these curves, and organizes their moduli through spaces of Schottky group parameters, enabling precise connections to tropical geometry and non-archimedean Hodge theory.

1. Non-Archimedean Schottky Groups and Their Actions

Let $k$ be a complete non-archimedean valued field and $\mathrm{P}^1_k$ the projective line over $k$. A non-archimedean Schottky group $\Gamma\subset\mathrm{PGL}_2(k)$ of rank $g\geq2$ is characterized as a free discrete subgroup generated by $g$ loxodromic elements, each with two distinct fixed points in $\mathrm{P}^1(k)$, and no nontrivial element of finite order. Such a group acts by Möbius transformations on the Berkovich projective line $\mathrm{P}^{1,\mathrm{an}}_k$, with orbits accumulating on a compact, perfect limit set $\Lambda_\Gamma\subset\mathrm{P}^1(k)$. The domain of discontinuity $k$0 is the maximal open locus on which $k$1 acts freely and properly discontinuously. These data are explicitly constructed from collections of disjoint open or closed Berkovich discs stable under the generators, leading to a fundamental domain $k$2 whose translates cover $k$3 (Poineau et al., 2020, Chambert-Loir et al., 2015).

2. Schottky Uniformization and Analytic Structure of Mumford Curves

For a Schottky group $k$4 as above, the quotient $k$5 inherits a $k$6-analytic curve structure, and indeed corresponds to a smooth proper algebraic curve (Mumford curve) of genus $k$7 after analytification. The analytic Schottky uniformization theorem states that every analytic Mumford curve arises (non-uniquely up to conjugacy) as such a quotient for a unique Schottky group, with the covering map $k$8 serving as the non-archimedean uniformization. The existence of a "good fundamental domain," formed by removing $k$9 carefully chosen disjoint open or closed Berkovich discs from $\mathrm{P}^1_k$0 corresponding to the generators and their inverses, is central to this construction, ensuring properness and the free action property (Poineau et al., 2020, Chambert-Loir et al., 2015).

The skeleton $\mathrm{P}^1_k$1 associated to the Berkovich analytic curve is a finite metric graph of genus $\mathrm{P}^1_k$2 whose edge lengths derive from the non-archimedean absolute value and the geometry of the group generators (Poineau et al., 2021).

3. Universal Non-Archimedean Schottky Space and Moduli

The moduli of non-archimedean Schottky groups, including the marking data specifying generators up to conjugacy and permutation, are parametrized by the Schottky space $\mathrm{P}^1_k$3, constructed as an open locus in non-archimedean analytic affine $\mathrm{P}^1_k$4-space with explicit coordinates ("Koebe coordinates") encoding the fixed points and multipliers of generators. Each point of $\mathrm{P}^1_k$5 determines a corresponding Schottky group and hence a Mumford curve. The universal Mumford curve $\mathrm{P}^1_k$6 forms a smooth proper family over $\mathrm{P}^1_k$7 via a universal Schottky action on the relative projective line, with the property that every non-archimedean Schottky uniformized (i.e., Mumford) curve arises as a fiber of this family (Poineau et al., 2021).

An induced right action of $\mathrm{P}^1_k$8, the outer automorphism group of the free group on $\mathrm{P}^1_k$9 generators, permutes markings and identifies isomorphic curves; the quotient $k$0 is the $k$1-analytic moduli space of genus $k$2 Mumford curves over $k$3, for any non-archimedean place $k$4 (Poineau et al., 2021).

4. Computational Theory: Non-Archimedean Theta Functions and Algorithms

The theory of non-archimedean theta functions underpins explicit computations in Schottky uniformization. For a non-archimedean Schottky group $k$5, the theta pairing $k$6 for degree zero divisors $k$7 on $k$8 is given by an infinite product of cross-ratios over the group elements: $k$9 where $\Gamma\subset\mathrm{PGL}_2(k)$0 denotes the usual cross-ratio. These pairings are $\Gamma\subset\mathrm{PGL}_2(k)$1-invariant, symmetric, and multiplicative, and generate the canonical coordinates and period matrix of the Jacobian of $\Gamma\subset\mathrm{PGL}_2(k)$2: $\Gamma\subset\mathrm{PGL}_2(k)$3 Efficient computation requires circumventing the exponential growth in group elements via an iterative, polynomial-time algorithm exploiting the free word structure of $\Gamma\subset\mathrm{PGL}_2(k)$4, affinoid algebra contraction mappings, and a careful decomposition by word-length and "tails." The core contraction property of the relevant operators ensures convergence and provides explicit complexity bounds: overall running time $\Gamma\subset\mathrm{PGL}_2(k)$5 for $\Gamma\subset\mathrm{PGL}_2(k)$6 the genus, $\Gamma\subset\mathrm{PGL}_2(k)$7 the $\Gamma\subset\mathrm{PGL}_2(k)$8-adic expansion length, and $\Gamma\subset\mathrm{PGL}_2(k)$9 the target precision (Masdeu et al., 2024).

5. Tropical and Geometric Group-Theoretic Correlates

Each Mumford curve $g\geq2$0 possesses a canonical skeleton $g\geq2$1, a metric graph reflecting the curve's reduction type. There exists a continuous surjective tropicalization map from Schottky space fibers $g\geq2$2 to the unprojectivized Culler–Vogtmann outer space $g\geq2$3 of marked metric graphs, encoding both the combinatorial and geometric data of degeneration. The further passage to the moduli space of tropical curves $g\geq2$4 is achieved by forgetting the marking. The natural "skeleton" retraction demonstrates that the Berkovich space $g\geq2$5 itself, as an analytic moduli, retracts to a polyhedral complex corresponding to $g\geq2$6, aligning the analytic, tropical, and group-theoretic perspectives (Poineau et al., 2021).

6. Model-Theoretic and Functional Transcendence Aspects

Non-archimedean Schottky uniformization admits a functional transcendence theory paralleling classical Ax–Lindemann results. For products of Mumford curves, every bi-algebraic irreducible subvariety of the uniformization domain $g\geq2$7 is characterized as "geodesic," defined by coordinate identifications via elements of $g\geq2$8 or by fixing coordinates. This structural rigidity is underpinned by analytic geometry of Berkovich spaces, group-theoretic stabilizer analysis, and $g\geq2$9-adic block-decomposition theorems (Chambert-Loir et al., 2015).

7. Extensions, Examples, and Limitations

Generalizations of the computational and uniformization framework apply to discontinuous groups containing Schottky subgroups of finite index, with the universal construction adapting via finite averaging. Construction of good fundamental domains from arbitrary generators has been algorithmically realized in polynomial time for certain classes (quaternionic Schottky groups) (Masdeu et al., 2024). The canonical example for genus $g$0 recovers Tate's uniformization of elliptic curves. For higher genera, explicit construction of generators and domains leads to explicit equations and geometric models of Mumford curves, with automorphism and skeleton structure computable from the Schottky data (Poineau et al., 2020, Poineau et al., 2021). The method is fundamentally obstructed for non-free (with torsion) discontinuous groups, where the geometry of orbits and the structure of the uniformized space can fail to yield a Mumford curve (Masdeu et al., 2024).

Non-archimedean Schottky uniformization thus equips the theory of $g$1-adic and valued-field algebraic curves with an analytic, group-theoretic moduli framework canonically compatible with tropicalization, outer automorphism action, and explicit computational models.