Cyclic Mumford Curves

• Cyclic Mumford curves are smooth projective algebraic curves defined over non-Archimedean fields that admit cyclic Galois covers and Schottky uniformization.
• They are characterized by explicit quadratic inequalities on branch data, ensuring totally degenerate reductions and a rigid analytic structure.
• Their moduli and invariants, expressed via p-adic theta functions, enable concrete classification and construction of examples across various genera.

A cyclic Mumford curve is a smooth projective algebraic curve defined over a non-Archimedean field, equipped with a Galois cover of the projective line by a cyclic group, and admitting a rigid-analytic uniformization by a Schottky group. These curves play a central role in $p$-adic and rigid analytic geometry, connecting the reduction theory of algebraic curves with explicit analytic and theta function constructions. Cyclic Mumford curves of degree $p$ are characterized both by combinatorial degeneration criteria in their Artin–Schreier or Kummer coverings and by explicit expressions for their moduli and branch-point invariants through $p$-adic theta functions.

1. Mumford Curves, Schottky Uniformization, and Cyclic Covers

Let $K$ be a complete, discretely valued (often algebraically closed) non-Archimedean field, with either characteristic $p > 0$ or characteristic $0$ (e.g., $K = \mathbb{C}_p$). A smooth, projective, geometrically connected $K$-curve $X$ of genus $g \geq 2$ is a Mumford curve if and only if it admits a Schottky uniformization: $$X^{\rm an} \cong (\mathbb{P}^1_K \setminus \mathcal{L})/\Gamma$$ where $\Gamma < \mathrm{PGL}_2(K)$ is a free, finitely generated, discrete, torsion-free subgroup, and $\mathcal{L} \subset \mathbb{P}^1(K)$ is the limit set of $\Gamma$. Equivalently, $X$ has totally degenerate split reduction: its stable model over $K^\circ$ has special fiber a union of $\mathbb{P}^1$'s meeting at $k$-rational ordinary double points (Mumford–Gerritzen–van der Put criterion).

A cyclic Mumford cover of degree $p$ is a finite Galois morphism: $\varphi: X \to \mathbb{P}^1_K$ of degree $p$ such that $X$ is a Mumford curve, and $\varphi$ is either an Artin–Schreier cover in positive characteristic ($p = \operatorname{char} K$), or a Kummer cover in characteristic zero, ramified at finitely many points.

In the cyclic context, the group of deck transformations is generated by an order-$p$ automorphism $\sigma_0$ operating on the ordinary region $\Omega$, so that $$\Gamma_0 = \langle \Gamma, \sigma_0 \rangle \subset \mathrm{PGL}_2(K)$$ acts discontinuously, $\Gamma$ is free of rank $g$, and $X = \Omega/\Gamma$ while $\Omega/\Gamma_0 \cong \mathbb{P}^1(K)$. The resulting Galois cover realizes the classical quotient structure with cyclic monodromy (Mikami, 2016, Kopeliovich, 30 Dec 2025).

2. Explicit Equations and Degeneration Criterion

For characteristic $p > 0$, after finite extension of scalars, a degree-$p$ cyclic cover $\varphi: X \to \mathbb{P}^1$ can always be given birationally as an Artin–Schreier cover: $$y^p - y = \sum_{i=1}^r \frac{\lambda_i}{x - a_i}, \qquad \lambda_i \in K^\times, \ a_i \in K, \ a_i \ne a_j$$ The genus of $X$ is $(p-1)(r-1)$, so $r \geq 3$ or $r=2$ with $p \geq 3$ ensures $g \geq 2$. The defining feature for $X$ to be a Mumford curve is a separation condition on the branch data: $$|\lambda_i \lambda_j| < |a_i - a_j|^2 \qquad \text{for all } i \ne j$$ This criterion ensures that the reductions of the local affinoid covers around the branch points remain (non-intersecting) unions of rational curves, so the special fiber is totally degenerate and $X$ is Schottky uniformizable (Mikami, 2016).

In characteristic zero (and tame degree $n$ not dividing the residue characteristic), cyclic coverings are given by Kummer equations: $y^n = c \prod_{i=1}^r \frac{x-a_i}{x-b_i}$ with a similar "smallness" hypothesis for $|c|$ to ensure split degenerate reduction (Mikami, 2016).

3. Rigid-Analytic and Combinatorial Proof Structure

The proof of the explicit criterion proceeds in two main analytic directions:

1. Sufficiency: If $|\lambda_i \lambda_j| < |a_i-a_j|^2$, then one constructs admissible affinoid coverings $\{U_n\}$ adapted to the branch points. Over each $U_n$ the cover reduces (after manipulations) to a local Artin–Schreier extension whose reduction is split. The local pieces glue to yield a totally degenerate special fiber, confirming that $X$ is a Mumford curve.
2. Necessity: If $X$ is a Mumford curve (admits Schottky uniformization), the $p$-adic tree of the Schottky group and the explicit expansion of local coordinates via group action yield the estimate $|\lambda_1 \lambda_2| < |a_1 - a_2|^2$. This argument generalizes via combinatorial analysis of mirrors of parabolic elements in the Bruhat–Tits tree, guaranteeing the separation for all pairs of branch points (Mikami, 2016).

This yields a valuation-adic classification: the only obstruction to being a degree-$p$ cyclic Mumford cover is the explicit system of quadratic inequalities among the $\lambda_i$ and $a_i$.

4. $p$-adic Schottky Theta Functions and $\lambda$-Invariants

For $K$ algebraically closed of characteristic zero, the uniformization framework gives rise to a $p$-adic period torus and associated theta functions. Writing $\Gamma \cong F_g$, let $$G_\Gamma = \operatorname{Hom}(\Gamma, K^\times) \cong (K^\times)^g$$ be the character torus, and $A_\Gamma \subset G_\Gamma$ the period lattice of multipliers.

The Schottky theta quotient for $a, b \in \Omega$ is: $$\Theta_{a,b}(z) = \prod_{\gamma \in \Gamma} \frac{z - \gamma a}{z - \gamma b}$$ with automorphy $$\Theta_{a,b}(\gamma z) = c_{a,b}(\gamma)\Theta_{a,b}(z)$$ for $c_{a,b}(\gamma) \in K^\times$. The analytic Jacobian is $J_\Gamma = G_\Gamma/A_\Gamma$, with a $p$-adic period matrix $\Omega = (\log q_{ij})$ from the multipliers.

A $p$-adic $\lambda$-function expressing the cross-ratio of branch points is constructed as a quotient of $p$-th powers of theta with characteristic: $$\Lambda(P) = \frac{\theta[D_1]_\Gamma(u_o(P))^p}{\theta[D_2]_\Gamma(u_o(P))^p}$$ where $D_1, D_2$ are non-special divisors of degree $g$ selecting ramification points, and $u_o$ is the Abel–Jacobi multiplicative map. The quotient of $\Lambda$ at two points realizes the classical cross-ratio: $$\frac{\Lambda(P_1)}{\Lambda(P_2)} = \frac{\theta[D_1]_\Gamma(u_o(P_1))^p\,\theta[D_2]_\Gamma(u_o(P_2))^p}{\theta[D_2]_\Gamma(u_o(P_1))^p\,\theta[D_1]_\Gamma(u_o(P_2))^p}$$ which equates, for ramification points $B_k,B_\ell$, to the rational formula

$$\frac{(a_k - a_\ell)(b_k - b_\ell)}{(a_k - b_\ell)(b_k - a_\ell)}$$

(Kopeliovich, 30 Dec 2025).

5. Moduli, Classification, and Explicit Examples

The quadratic inequalities $|\lambda_i\lambda_j|<|a_i - a_j|^2$ cut out the moduli space of degree-$p$ cyclic Mumford covers inside the Artin–Schreier family. This results in an explicit, valuation-theoretic parameterization of these covers.

In the hyperelliptic case ($p=2$), these constructions recover the classical Tate curve theory and the expressions for the cross-ratio ("$\lambda$") as quotients of even Riemann theta constants, paralleling the Thomae formula. For genus $g=2$ and arbitrary $p$, explicit power-series (theta expansion) expressions of the moduli invariant $\lambda$ in the entries of the $p$-adic period matrix $\Omega$ are available, generalizing Teitelbaum's formulas (Kopeliovich, 30 Dec 2025).

Concrete examples are generated by choosing branch points $a_i$ with valuations forming an arithmetic progression and residues $\lambda_i$ of prescribed valuation just below $|a_i - a_j|$, yielding one-parameter families of cyclic Mumford curves of arbitrarily large genus (Mikami, 2016).

6. Comparison with Complex Theory and Broader Context

Classically, cross-ratios of branch points for cyclic covers are expressed via Riemann theta functions on complex Jacobians, with the Thomae and Schottky–Jung formulas connecting even theta constants to moduli invariants. In the $p$-adic context, the theta function construction is fully parallel: the sum over period lattices is replaced by $p$-adic convergent series, and the exponential is modeled by a multiplicative bilinear pairing, yet preserves the combinatorial relation among characteristics and cross-ratios. This suggests a precise $p$-adic analogue of the classical analytic theory interpolating between geometry, automorphy, and moduli (Kopeliovich, 30 Dec 2025).

7. Key Theorems and Results

Main classification for positive characteristic—due to Mikami (Mikami, 2016):

• $X$ given by $y^p - y = \sum \lambda_i/(x-a_i)$ is a Mumford curve iff $|\lambda_i \lambda_j| < |a_i - a_j|^2$ for all $i \ne j$.
• The moduli space for such covers is determined by these quadratic inequalities.
• For $p=2$ or $r=2,g=p-1$, the results encompass classical Tate curve constructions and the split Artin–Schreier case.

Main analytic-theta function framework—following Kopeliovich (Kopeliovich, 30 Dec 2025):

• The branch-point cross-ratio for cyclic Mumford covers is expressed as a quotient of $p$-adic theta functions evaluated at the $p$-adic period matrix.
• Moduli invariants are encoded via explicit power series in the period matrix, making analysis of $p$-adic variation of moduli possible in all genera.

These results comprise a comprehensive and explicit description of cyclic Mumford curves, bridging rigid analytic geometry, Galois theory of curves, and $p$-adic special functions.