Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

$p$-Adic $λ$ Functions for Cyclic Mumford Curve

Yaacov Kopeliovich

TL;DR

This work provides a $p$-adic analogue of classical cross-ratio formulas for the branch points of cyclic and hyperelliptic curves by embedding them into the framework of $p$-adic theta functions on Mumford curves. The authors develop a comprehensive machinery: (i) a dimension theory for non-special divisors on cyclic covers in the $p$-adic setting; (ii) explicit transformation laws for $p$-adic theta functions under the normalizer action of the Schottky group and the computation of branch-point images in the analytic Jacobian; and (iii) a $p$-adic Teitelbaum lambda function built from theta functions with characteristics that yields cross-ratio expressions for branch points, including a hyperelliptic specialization at $p=2$. The results unify and extend known complex-analytic formulas to the non-Archimedean realm, enabling cross-ratio calculations and period-lambda relations on a broader class of Mumford curves and suggesting avenues to generalize to all Galois Mumford covers. The work thus advances the toolkit for understanding $p$-adic moduli of branch points and their period data in a robust, theta-function framework.

Abstract

We express the branch points cross ratio of cyclic Mumford curves as quotients of $p$-adic theta functions evaluated at the p-adic period matrix

$p$-Adic $λ$ Functions for Cyclic Mumford Curve

TL;DR

This work provides a -adic analogue of classical cross-ratio formulas for the branch points of cyclic and hyperelliptic curves by embedding them into the framework of -adic theta functions on Mumford curves. The authors develop a comprehensive machinery: (i) a dimension theory for non-special divisors on cyclic covers in the -adic setting; (ii) explicit transformation laws for -adic theta functions under the normalizer action of the Schottky group and the computation of branch-point images in the analytic Jacobian; and (iii) a -adic Teitelbaum lambda function built from theta functions with characteristics that yields cross-ratio expressions for branch points, including a hyperelliptic specialization at . The results unify and extend known complex-analytic formulas to the non-Archimedean realm, enabling cross-ratio calculations and period-lambda relations on a broader class of Mumford curves and suggesting avenues to generalize to all Galois Mumford covers. The work thus advances the toolkit for understanding -adic moduli of branch points and their period data in a robust, theta-function framework.

Abstract

We express the branch points cross ratio of cyclic Mumford curves as quotients of -adic theta functions evaluated at the p-adic period matrix
Paper Structure (11 sections, 23 theorems, 69 equations)

This paper contains 11 sections, 23 theorems, 69 equations.

Table of Contents

  1. introduction
  2. Non-Special Divisors
  3. Cyclic Mumford Curves
  4. Analytical Jacobian Variety of Mumford curves
  5. $p$-Adic Theta Functions
  6. Images of branch points in the Jacobian variety
  7. Non-Special Divisors
  8. $\lambda$ Teitelbaum functions for Mumford cyclic curves
  9. Hyperelliptic case
  10. $p$-adic Lambda function for hyper-elliptic curves
  11. conclusion

Key Result

Theorem 2.1

Let $D\,{=}\,\sum_{i=1}^n d_i D_i$ be a divisor supported on finite branch points, and $d_j \,{\in}\, \{0,\dots,p-1\}$. Define the set $\{g_k\}_{k=0}^{p-1}$ by the equations where $\overline{r}$ denotes the remainder of the division of $r$ by $p$. Then

Theorems & Definitions (47)

  • Definition 2.1
  • Theorem 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 3.1
  • Proposition 3.1
  • Theorem 3.2
  • Definition 3.2
  • Theorem 3.3
  • ...and 37 more