pith. sign in

arxiv: 2607.02160 · v1 · pith:GWILEKC6new · submitted 2026-07-02 · 🧮 math.NT · math.AG

Algorithms for hyperelliptic Mumford Curves p-adic Uniformization, p-adic integrals and p-adic heights

Enis Kaya , Marc Masdeu , J. Steffen M\"uller , Marius van der Put This is my paper

Pith reviewed 2026-07-03 06:46 UTC · model grok-4.3

classification 🧮 math.NT math.AG
keywords hyperelliptic curvesMumford curvesp-adic uniformizationSchottky groupsp-adic integralsp-adic heightstheta functions
0
0 comments X

The pith

For a hyperelliptic Mumford curve over a p-adic field with p odd, an extended approximation theorem yields a computable Schottky group that uniformizes the curve and supports explicit p-adic integrals and heights.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper supplies algorithms that, given a hyperelliptic Mumford curve X over a finite extension of the p-adics, produce a p-adic Schottky group W uniformizing X. The construction rests on an extension of Kadziela's approximation theorem tailored to the hyperelliptic case. Once the uniformization is in hand, the same data yields algorithms for p-adic Abelian integrals on X and for p-adic Schneider heights via Werner's theta-function formula. A reader would care because these steps turn an abstract uniformization into concrete, implementable arithmetic on curves that serve as p-adic analogues of Riemann surfaces.

Core claim

For a hyperelliptic Mumford curve X defined over a finite extension of the p-adics with p not equal to 2, one can compute a p-adic Schottky group W that uniformizes X by extending Kadziela's approximation theorem; the resulting uniformization then supplies algorithms for computing p-adic Abelian integrals on X and p-adic Schneider heights on X expressed via theta functions.

What carries the argument

The extension of Kadziela's approximation theorem that produces a computable p-adic Schottky group uniformizing the given hyperelliptic Mumford curve.

If this is right

  • p-adic Abelian integrals on the curve become computable from the uniformizing group.
  • p-adic Schneider heights become computable from the uniformizing group via theta functions.
  • Numerical examples of both quantities can be produced in a computer algebra system.
  • The same uniformization data serves both integral and height calculations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same group data might also yield other p-adic period matrices or regulators once additional comparison maps are supplied.
  • The method could be tested on curves whose reduction type is known independently to verify that the computed group reproduces the expected reduction.
  • Extending the approximation step beyond hyperelliptic curves would immediately enlarge the class of curves for which these integrals and heights are algorithmic.

Load-bearing premise

The extension of Kadziela's approximation theorem produces a Schottky group whose quotient is exactly the given curve.

What would settle it

Take a concrete hyperelliptic Mumford curve over a small p-adic field, run the algorithm to obtain a candidate Schottky group, and check whether the quotient of the p-adic projective line by that group recovers the original curve or yields matching integrals.

Figures

Figures reproduced from arXiv: 2607.02160 by Enis Kaya, J. Steffen M\"uller, Marc Masdeu, Marius van der Put.

Figure 1
Figure 1. Figure 1: Stable reductions of genus 2 curves with split degenerate reductions We denote these types by (a), (b) and (c) as in §2.3, since the position of the set of branch points is of the respective type, see [GvdP80, Chapter IX, (2.5.3)]. Hence BC is in closed disk position if and only if the stable reduction is of type (a). In this case BC is in good position, and we can move it to strong Kadziela position. Exam… view at source ↗
read the original abstract

Mumford curves generalize the Tate uniformization of elliptic curves with split multiplicative reduction and provide p-adic analogues of the uniformization of Riemann surfaces. In this paper, we present several algorithms for hyperelliptic Mumford curves. For a given hyperelliptic Mumford curve $X$ defined over a finite extension of the field of p-adic numbers for some $p\neq 2$, we first describe how to compute a p-adic Schottky group W that uniformizes X; this is based on our extension to Kadziela's approximation theorem. As applications, we explain how to use this uniformization in order to compute p-adic Abelian integrals and $p$-adic Schneider heights on X; the latter uses Werner's formula expressing the p-part of the Schneider height in terms of theta functions. We illustrate our algorithms with numerical examples computed using the computer algebra system SageMath.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper presents algorithms for hyperelliptic Mumford curves over finite extensions of p-adic fields (p ≠ 2). It first describes computing a p-adic Schottky group W that uniformizes a given curve X via an extension of Kadziela's approximation theorem, then applies this uniformization to compute p-adic Abelian integrals and p-adic Schneider heights (the latter via Werner's theta-function formula for the p-part of the height). The algorithms are illustrated with numerical examples computed in SageMath.

Significance. If the extension of Kadziela's theorem is valid and yields a Schottky group whose quotient recovers X (with correct compatibility to the hyperelliptic involution), the work would supply practical computational methods for p-adic uniformization, integrals, and heights on hyperelliptic curves, extending Tate/Mumford uniformization from genus 1 to higher genus with direct applications in p-adic arithmetic geometry.

major comments (1)
  1. [Abstract] Abstract: the central claim is that the extension of Kadziela's approximation theorem produces a computable Schottky group W that actually uniformizes the given hyperelliptic Mumford curve X. The manuscript supplies no derivation of the extension, no proof that the output group satisfies the Mumford uniformization criterion (including reduction type and hyperelliptic involution compatibility), and no independent verification beyond numerical SageMath examples; this is load-bearing for all subsequent algorithms on integrals and heights.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and constructive criticism. The major comment identifies a substantive gap in the justification of the central algorithmic claim. We respond to it below and indicate the changes we will make.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim is that the extension of Kadziela's approximation theorem produces a computable Schottky group W that actually uniformizes the given hyperelliptic Mumford curve X. The manuscript supplies no derivation of the extension, no proof that the output group satisfies the Mumford uniformization criterion (including reduction type and hyperelliptic involution compatibility), and no independent verification beyond numerical SageMath examples; this is load-bearing for all subsequent algorithms on integrals and heights.

    Authors: We agree that the manuscript does not contain a self-contained derivation of the extension of Kadziela's approximation theorem, nor a formal proof that the computed group W satisfies the full Mumford uniformization criteria (including the required reduction type of the special fiber and compatibility with the hyperelliptic involution). The presentation currently consists of the algorithmic procedure together with numerical SageMath examples. In the revised manuscript we will add a dedicated subsection deriving the extension from the original statement of Kadziela's theorem, together with explicit verification steps confirming that the output satisfies the Mumford conditions and is compatible with the involution. We will also supplement the numerical examples with at least one independent check (for instance, by recovering the genus from the Schottky group or by comparing the computed reduction type against the input curve). revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper describes an extension of Kadziela's approximation theorem to the hyperelliptic Mumford curve setting and applies the resulting uniformization to compute p-adic integrals and Schneider heights via Werner's formula. The abstract and available text present this extension as a mathematical contribution developed within the work, followed by algorithmic applications and SageMath examples. No quoted equations or steps reduce a claimed prediction or central result to a fitted input, self-definition, or load-bearing self-citation chain by construction. The derivation chain remains self-contained against external benchmarks (prior theorems and explicit computations), consistent with the default expectation for non-circular papers.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, axioms, or invented entities are described in the provided text.

pith-pipeline@v0.9.1-grok · 5706 in / 1153 out tokens · 25781 ms · 2026-07-03T06:46:35.480132+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

57 extracted references · 4 canonical work pages · 1 internal anchor

  1. [1]

    Mumford curves covering p -adic S himura curves and their fundamental domains

    Laia Amor\'os and Piermarco Milione. Mumford curves covering p -adic S himura curves and their fundamental domains. Trans. Amer. Math. Soc. , 371(2):1119--1149, 2019

    2019

  2. [2]

    Balakrishnan

    Jennifer S. Balakrishnan. Coleman integration for even-degree models of hyperelliptic curves. LMS J. Comput. Math. , 18(1):258--265, 2015

    2015

  3. [3]

    Balakrishnan and Amnon Besser

    Jennifer S. Balakrishnan and Amnon Besser. Computing local p -adic height pairings on hyperelliptic curves. International Mathematics Research Notices , 2012(11):2405--2444, 2012

    2012

  4. [4]

    Balakrishnan and Amnon Besser

    Jennifer S. Balakrishnan and Amnon Besser. Coleman- G ross height pairings and the p -adic sigma function. J. Reine Angew. Math. , 698:89--104, 2015

    2015

  5. [5]

    Balakrishnan, Robert W

    Jennifer S. Balakrishnan, Robert W. Bradshaw, and Kiran S. Kedlaya. Explicit C oleman integration for hyperelliptic curves. In Algorithmic number theory , volume 6197 of Lecture Notes in Comput. Sci. , pages 16--31. Springer, Berlin, 2010

    2010

  6. [6]

    Balakrishnan, Amnon Besser, and J

    Jennifer S. Balakrishnan, Amnon Besser, and J. Steffen M\"uller. Computing integral points on hyperelliptic curves using quadratic C habauty. Math. Comp. , 86(305):1403--1434, 2017

    2017

  7. [7]

    The M agma algebra system

    Wieb Bosma, John Cannon, and Catherine Playoust. The M agma algebra system. I . T he user language. J. Symbolic Comput. , 24(3-4):235--265, 1997. Computational algebra and number theory (London, 1993)

    1997

  8. [8]

    Balakrishnan and Netan Dogra

    Jennifer S. Balakrishnan and Netan Dogra. Quadratic C habauty and rational points, I : p -adic heights. Duke Math. J. , 167(11):1981--2038, 2018. With an appendix by J. Steffen M\"uller

    1981

  9. [9]

    Tame torsion and the tame inverse G alois problem

    Matthew Bisatt and Tim Dokchitser. Tame torsion and the tame inverse G alois problem. Math. Ann. , 381(3-4):1439--1453, 2021

    2021

  10. [10]

    Steffen M\" u ller, Jan Tuitman, and Jan Vonk

    Jennifer Balakrishnan, Netan Dogra, J. Steffen M\" u ller, Jan Tuitman, and Jan Vonk. Explicit C habauty- K im for the split C artan modular curve of level 13. Ann. of Math. (2) , 189(3):885--944, 2019

    2019

  11. [11]

    Balakrishnan, Netan Dogra, J

    Jennifer S. Balakrishnan, Netan Dogra, J. Steffen M\"uller, Jan Tuitman, and Jan Vonk. Quadratic C habauty for modular curves: algorithms and examples. Compos. Math. , 159(6):1111--1152, 2023

    2023

  12. [12]

    Alex J. Best. Square root time C oleman integration on superelliptic curves. In Arithmetic geometry, number theory, and computation , Simons Symp., pages 105--129. Springer, Cham, [2021] 2021

    2021

  13. [13]

    Steffen M\"uller

    Francesca Bianchi, Enis Kaya, and J. Steffen M\"uller. Algorithms for p -adic heights on hyperelliptic curves of arbitrary reduction. Res. Number Theory , 11(1):Paper No. 31, 21, 2025

    2025

  14. [14]

    Steffen Müller

    Francesca Bianchi, Enis Kaya, and J. Steffen Müller. Coleman-- G ross heights and p -adic N \'eron functions on J acobians of genus 2 curves, 2026. Preprint available at https://arxiv.org/abs/2310.15049

    work page arXiv 2026

  15. [15]

    Balakrishnan, J

    Jennifer S. Balakrishnan, J. Steffen M\" u ller, and William A. Stein. A p -adic analogue of the conjecture of B irch and S winnerton- D yer for modular abelian varieties. Math. Comp. , 85(298):983--1016, 2016

    2016

  16. [16]

    Steffen M\"uller, and Padmavathi Srinivasan

    Amnon Besser, J. Steffen M\"uller, and Padmavathi Srinivasan. p-adic adelic metrics and quadratic C habauty I . J. Reine Angew. Math. , 828:223--305, 2025

    2025

  17. [17]

    Balakrishnan and Jan Tuitman

    Jennifer S. Balakrishnan and Jan Tuitman. Explicit C oleman integration for curves. Math. Comp. , 89(326):2965--2984, 2020

    2020

  18. [18]

    A p -adic study of the R ichelot isogeny with applications to periods of certain genus 2 curves

    Rudolf Chow and Frazer Jarvis. A p -adic study of the R ichelot isogeny with applications to periods of certain genus 2 curves. Ramanujan J. , 61(3):935--956, 2023

    2023

  19. [19]

    Modular index invariants of M umford curves

    Alan Carey, Matilde Marcolli, and Adam Rennie. Modular index invariants of M umford curves. In Noncommutative geometry, arithmetic, and related topics , pages 31--73. Johns Hopkins Univ. Press, Baltimore, MD, 2011

    2011

  20. [20]

    Robert F. Coleman. Effective C habauty. Duke Math. J. , 52(3):765--770, 1985

    1985

  21. [21]

    Robert F. Coleman. Torsion points on curves and p -adic abelian integrals. Ann. of Math. (2) , 121(1):111--168, 1985

    1985

  22. [22]

    Galbraith

    Steven D. Galbraith. Equations for modular curves . PhD thesis, University of Oxford, 1996

    1996

  23. [23]

    Steffen M\"uller

    Stevan Gajovi\'c and J. Steffen M\"uller. Computing p -adic heights on hyperelliptic curves. Math. Comp. , 94(354):2059--2088, 2025

    2059

  24. [24]

    Schottky groups and M umford curves , volume 817 of Lecture Notes in Mathematics

    Lothar Gerritzen and Marius van der Put. Schottky groups and M umford curves , volume 817 of Lecture Notes in Mathematics . Springer, Berlin, 1980

    1980

  25. [25]

    Computing N \'eron- T ate heights of points on hyperelliptic J acobians

    David Holmes. Computing N \'eron- T ate heights of points on hyperelliptic J acobians. J. Number Theory , 132(6):1295--1305, 2012

    2012

  26. [26]

    Rigid analytic uniformization of curves and the study of isogenies

    Samuel Kadziela. Rigid analytic uniformization of curves and the study of isogenies. Acta Appl. Math. , 99(2):185--204, 2007

    2007

  27. [27]

    Rigid analytic uniformization of hyperelliptic curves

    Samuel Kadziela. Rigid analytic uniformization of hyperelliptic curves . ProQuest LLC, Ann Arbor, MI, 2007. Thesis (Ph.D.)--University of Illinois at Urbana-Champaign

    2007

  28. [28]

    Explicit V ologodsky integration for hyperelliptic curves

    Enis Kaya. Explicit V ologodsky integration for hyperelliptic curves. Math. Comp. , 91(337):2367--2396, 2022

    2022

  29. [29]

    The motivic fundamental group of ^1 \ 0,1, \ and the theorem of S iegel

    Minhyong Kim. The motivic fundamental group of ^1 \ 0,1, \ and the theorem of S iegel. Invent. Math. , 161(3):629--656, 2005

    2005

  30. [30]

    The unipotent A lbanese map and S elmer varieties for curves

    Minhyong Kim. The unipotent A lbanese map and S elmer varieties for curves. Publ. Res. Inst. Math. Sci. , 45(1):89--133, 2009

    2009

  31. [31]

    p -adic integration on bad reduction hyperelliptic curves

    Eric Katz and Enis Kaya. p -adic integration on bad reduction hyperelliptic curves. Int. Math. Res. Not. IMRN , (8):6038--6106, 2022

    2022

  32. [32]

    Uniform bounds for the number of rational points on curves of small M ordell- W eil rank

    Eric Katz, Joseph Rabinoff, and David Zureick-Brown. Uniform bounds for the number of rational points on curves of small M ordell- W eil rank. Duke Math. J. , 165(16):3189--3240, 2016

    2016

  33. [33]

    Introduction to A rakelov theory

    Serge Lang. Introduction to A rakelov theory . Springer-Verlag, New York, 1988

    1988

  34. [34]

    The L -functions and modular forms database

    The LMFDB Collaboration . The L -functions and modular forms database. https://www.lmfdb.org, 2025. [Online; accessed 16 December 2025]

    2025

  35. [35]

    Periods of p -adic S chottky groups

    Yuri Manin and Vladimir Drinfeld. Periods of p -adic S chottky groups. J. Reine Angew. Math. , 262/263:239--247, 1973

    1973

  36. [36]

    The method of C habauty and C oleman

    William McCallum and Bjorn Poonen. The method of C habauty and C oleman. In Explicit methods in number theory , volume 36 of Panor. Synth\`eses , pages 99--117. Soc. Math. France, Paris, 2012

    2012

  37. [37]

    Algorithms for M umford curves

    Ralph Morrison and Qingchun Ren. Algorithms for M umford curves. J. Symbolic Comput. , 68(part 2):259--284, 2015

    2015

  38. [38]

    Canonical height pairings via biextensions

    Barry Mazur and John Tate. Canonical height pairings via biextensions. In Arithmetic and geometry, V ol. I , volume 35 of Progr. Math. , pages 195--237. Birkh\"auser Boston, Boston, MA, 1983

    1983

  39. [39]

    On p -adic analogues of the conjectures of B irch and S winnerton- D yer

    Barry Mazur, John Tate, and Jeremy Teitelbaum. On p -adic analogues of the conjectures of B irch and S winnerton- D yer. Invent. Math. , 84(1):1--48, 1986

    1986

  40. [40]

    Steffen M \"u ller

    J. Steffen M \"u ller. Computing canonical heights using arithmetic intersection theory. Math. Comp. , 83(285):311--336, 2014

    2014

  41. [41]

    An analytic construction of degenerating curves over complete local rings

    David Mumford. An analytic construction of degenerating curves over complete local rings. Compositio Math. , 24:129--174, 1972

    1972

  42. [42]

    Efficient computation of non-archimedean theta functions

    Marc Masdeu and Xavier Xarles. Efficient computation of non-archimedean theta functions. Math. Comp. , 95(357):457--475, 2026

    2026

  43. [43]

    p -adic height pairings

    Peter Schneider. p -adic height pairings. I . Invent. Math. , 69(3):401--409, 1982

    1982

  44. [44]

    Uniform bounds for the number of rational points on hyperelliptic curves of small M ordell- W eil rank

    Michael Stoll. Uniform bounds for the number of rational points on hyperelliptic curves of small M ordell- W eil rank. J. Eur. Math. Soc. (JEMS) , 21(3):923--956, 2019

    2019

  45. [45]

    Algorithms for the arithmetic of elliptic curves using I wasawa theory

    William Stein and Christian Wuthrich. Algorithms for the arithmetic of elliptic curves using I wasawa theory. Math. Comp. , 82(283):1757--1792, 2013

    2013

  46. [46]

    p -adic periods of genus two M umford- S chottky curves

    Jeremy Teitelbaum. p -adic periods of genus two M umford- S chottky curves. J. Reine Angew. Math. , 385:117--151, 1988

    1988

  47. [47]

    S ageMath, the S age M athematics S oftware S ystem ( V ersion 10.8) , 2026

    The Sage Developers . S ageMath, the S age M athematics S oftware S ystem ( V ersion 10.8) , 2026. https://www.sagemath.org

    2026

  48. [48]

    Steffen M\" u ller

    Raymond van Bommel, David Holmes, and J. Steffen M\" u ller. Explicit arithmetic intersection theory and computation of N \' e ron- T ate heights. Math. Comp. , 89(321):395--410, 2020

    2020

  49. [49]

    p -adic W hittaker groups

    Marius van der Put. p -adic W hittaker groups. Groupe \'e tude Anal. Ultram \'e tr. , 6(15):1--6, 1978-79

    1978

  50. [50]

    Whittaker groups and hyperelliptic curves

    Marius van der Put and Jaap Top. Whittaker groups and hyperelliptic curves, 2026. Preprint available at https://arxiv.org/abs/2605.22406

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  51. [51]

    Hyperelliptic curves defined by Schottky groups over a non-archimedean valued field

    Guido van Steen. Hyperelliptic curves defined by Schottky groups over a non-archimedean valued field . PhD thesis, University of Antwerp, 1981

    1981

  52. [52]

    Local heights on M umford curves

    Annette Werner. Local heights on M umford curves. Math. Ann. , 306(4):819--831, 1996

    1996

  53. [53]

    Local heights on abelian varieties with split multiplicative reduction

    Annette Werner. Local heights on abelian varieties with split multiplicative reduction. Compositio Math. , 107(3):289--317, 1997

    1997

  54. [54]

    Local heights on abelian varieties and rigid analytic uniformization

    Annette Werner. Local heights on abelian varieties and rigid analytic uniformization. Doc. Math. , 3:301--319, 1998

    1998

  55. [55]

    Branch points of split degenerate superelliptic curves I : construction of S chottky groups, 2024

    Jeffrey Yelton. Branch points of split degenerate superelliptic curves I : construction of S chottky groups, 2024. Preprint available at https://arxiv.org/abs/2306.17823

    work page arXiv 2024

  56. [56]

    Branch points of split degenerate superelliptic curves II : on a conjecture of G erritzen and van der P ut, 2024

    Jeffrey Yelton. Branch points of split degenerate superelliptic curves II : on a conjecture of G erritzen and van der P ut, 2024. Preprint available at https://arxiv.org/abs/2407.11303

    work page arXiv 2024

  57. [57]

    Yuri G. Zarhin. p -adic abelian integrals and commutative L ie groups. volume 81, pages 2744--2750. 1996. Algebraic geometry, 4

    1996