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arxiv: 2605.22406 · v1 · pith:MH4W5I2Xnew · submitted 2026-05-21 · 🧮 math.AG

Whittaker groups and hyperelliptic curves

Marius van der Put , Jaap Top This is my paper

Pith reviewed 2026-05-22 02:26 UTC · model grok-4.3

classification 🧮 math.AG
keywords Whittaker groupshyperelliptic Mumford curvesSchottky groupsrigid etale coveringsfixed pointsbranch locustheta functionsp-adic periods
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The pith

The map from good fixed-point configurations to branch loci on hyperelliptic Mumford curves is a rigid etale covering with Galois group a product of copies of plus or minus one.

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

The paper defines Whittaker groups as discontinuous subgroups of PGL(2,K) freely generated by involutions whose fixed points lie in good position over a complete non-archimedean field K with residue characteristic not two. Index-two Schottky subgroups inside such a group produce hyperelliptic Mumford curves of genus g whose branch locus lies in the projective line. For any configuration of g+1 pairs of points, the authors introduce rigid spaces of allowable fixed points and of corresponding branch points, then prove that the natural morphism between these spaces is an etale covering whose Galois group is of the form {±1}^{d-1} for some d. This relation supplies explicit theta-function expressions for the branch points, periods, and heights of the resulting curves. The claim is checked by direct approximation in all cases of genus two and three.

Core claim

A Whittaker group G is freely generated by g+1 involutions in PGL(2,K) whose fixed points are placed in good position. An index-two Schottky subgroup W yields a hyperelliptic Mumford curve Omega/W of genus g with branch locus B. For a configuration (P,m) of g+1 pairs of points, the rigid space Fix_{P,m} of fixed points in good position maps via a natural morphism FB to the rigid space Branch_{P,m} of branch points; this morphism is a rigid etale covering whose Galois group is {±1}^{d-1} for some positive integer d. The same data also yields explicit theta-function formulas for the branch points and p-adic periods of the curve.

What carries the argument

The rigid etale covering morphism FB: Fix_{P,m} → Branch_{P,m} that relates configurations of fixed points in good position to the branch locus of the associated Whittaker curve.

If this is right

  • Branch points and p-adic periods of the curves can be written directly as values of theta functions attached to the groups.
  • Analytic reductions of Whittaker curves admit a classification in terms of the generators and their fixed-point data.
  • The relation between fixed points and branch loci holds for all genera, with explicit verification already completed for g=2 and g=3.
  • The construction supplies a concrete parametrization of all such hyperelliptic curves arising from Whittaker groups.

Where Pith is reading between the lines

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

  • The etale covering may induce a correspondence between moduli spaces of point-pair configurations and moduli of hyperelliptic curves over the non-archimedean field.
  • One could test the result further by determining the precise degree d as a function of the genus or by examining ramification loci in the rigid spaces.
  • The same covering structure might extend to other uniformizations or to curves of higher genus that admit similar involution generators.

Load-bearing premise

The fixed points of the generating involutions must lie in good position in order to define the Whittaker group and to equip the configuration with the rigid spaces Fix and Branch on which the covering morphism is defined.

What would settle it

An explicit computation or counter-example in genus four or higher showing that FB fails to be etale or possesses a Galois group other than a product of copies of {±1} would disprove the central claim.

Figures

Figures reproduced from arXiv: 2605.22406 by Jaap Top, Marius van der Put.

Figure 1
Figure 1. Figure 1: The three configurations for fixed points for Whittaker curves of genus 2, here represented as pointed B-trees, and their generalizations of higher genus are studied in detail in the Subsections 5.3.1–5.3.3. 5.3. Examples of configurations (P, m), the moduli spaces F ixP,m, BranchP,m and the morphism F B. 5.3.1. Case (a). This case, already treated in Example 5.2, is an example of the “closed disk conditio… view at source ↗
Figure 2
Figure 2. Figure 2: All configurations of genus three. The number of branch points that are mapped to a vertex is indicated by ∅, 1 or 2. follows. The branch data consists of tuples {a0, b0}, · · · , {a3, b3} modulo the simultaneous action of PGL2. The tuples are normalized by imposing the conditions a0 = 0, a1 = 1, a3 = ∞. The rigid space Branch(a,g=3) consists of the tuples (B0, T1, T2, A, B3) ∈ K5 given by the (in)equaliti… view at source ↗
read the original abstract

Let K be a complete, non-archimedean valued field with a residue field of characteristic different from 2. A Whittaker group G is a discontinuous subgroup of PGL(2,K), freely generated by elements s_0,...,s_g of order two, each defined by a pair of fixed points {a_0,b_0},...,{a_g,b_g}. These fixed points are called ``in good position''. A subgroup W in G of index 2 is a Schottky group and produces a hyperelliptic Mumford curve Omega/W --> Omega/G = P^1, called `Whittaker curve', of genus g and with branch locus B in P^1(K). An explicit parametrization of Whittaker curves in terms of theta functions for W and G and the data of the fixed points, is developed. In particular, this allows one to express the branched points (and other data such as p-adic periods and p-adic heights) in terms of values of theta functions. A central theme of this paper is the relation between the fixed points and the branch locus. For a given configuration (P,m) of $g+1$ pairs of points in P^1, one defines a rigid space Fix_{P,m} of fixed points in good position with that configuration and a rigid space of branched points $ Branch_{P,m} in that configuration. A main result is that the natural morphism FB: Fix_{P,m} --> Branch_{P,m} is a rigid etale covering with Galois group {\pm 1}^{d-1} for some d>0. For all cases of genus g=2,3 (and for some more), an approximation of FB is computed which confirms the main result. Classification of Whittaker groups and analytic reductions of Whittaker curves is another important issue of this paper. The background material in this paper complements the work of L.~Gerritzen, G.~Van Steen, F.~Herrlich and others. It involves re-examination of some proofs, the derivation of properties of semi-stable analytic reductions and studying good position of fixed points.

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 / 2 minor

Summary. The manuscript defines Whittaker groups as discontinuous subgroups of PGL(2,K) freely generated by involutions s_0,...,s_g of order two whose fixed points lie in good position. It constructs associated hyperelliptic Mumford curves via index-2 Schottky subgroups W, develops explicit theta-function parametrizations of the curves and their branch loci, and introduces rigid spaces Fix_{P,m} and Branch_{P,m} for a configuration (P,m) of g+1 point pairs. The central claim is that the natural morphism FB: Fix_{P,m} → Branch_{P,m} is a rigid étale covering with Galois group {±1}^{d-1}. Low-genus cases (g=2,3) are checked by direct approximation; the paper also classifies Whittaker groups, studies analytic reductions, and re-derives semi-stable reduction properties from prior work of Gerritzen–Van Steen–Herrlich.

Significance. If the étale-covering result holds, the work supplies an explicit dictionary between fixed-point configurations and branch loci for p-adic hyperelliptic curves, together with concrete expressions for branch points, periods, and heights in terms of theta functions. The low-genus verifications and the re-examination of good-position and semi-stable reduction properties add concrete computational content and strengthen the foundational literature on rigid analytic Mumford curves.

major comments (1)
  1. [section stating the main result on FB (and the low-genus approximation subsection)] The central claim that FB is a rigid étale covering with Galois group {±1}^{d-1} is load-bearing for the paper's main theorem. The construction begins from the external definitions of PGL(2,K) and rigid spaces and derives the étale property from the theta-function relations and good-position assumption, yet the manuscript supplies neither a complete general proof of étaleness nor quantitative error bounds on the low-genus approximations that are used to confirm the Galois group. This gap directly affects verifiability of the principal result.
minor comments (2)
  1. [introductory definitions] The notion of 'good position' for the fixed-point pairs is invoked repeatedly to define both the Whittaker group and the rigid spaces, but it is not isolated as a numbered definition or axiom; a formal statement would improve readability.
  2. [background material] Citations to the background results of Gerritzen–Van Steen–Herrlich are present but could be made more precise by referencing specific lemmas or theorems rather than the authors' names alone.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thorough review and insightful comments on our paper. We address the major concern regarding the proof of the central result on the rigid étale covering below.

read point-by-point responses

  1. Referee: [section stating the main result on FB (and the low-genus approximation subsection)] The central claim that FB is a rigid étale covering with Galois group {±1}^{d-1} is load-bearing for the paper's main theorem. The construction begins from the external definitions of PGL(2,K) and rigid spaces and derives the étale property from the theta-function relations and good-position assumption, yet the manuscript supplies neither a complete general proof of étaleness nor quantitative error bounds on the low-genus approximations that are used to confirm the Galois group. This gap directly affects verifiability of the principal result.

    Authors: The manuscript derives the étale property of FB from the explicit theta-function relations under the good-position assumption, as developed in the section on the morphism FB and the parametrizations of Whittaker curves. We agree that expanding the steps of this derivation would improve verifiability for readers. In the revised manuscript we will provide a more detailed general proof of étaleness, including explicit invocation of the relevant theta identities. We will also supply quantitative error bounds for the low-genus approximations used to confirm the Galois group. These additions will appear in the section stating the main result on FB. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from external definitions

full rationale

The paper starts from the standard external definition of PGL(2,K) and rigid analytic spaces over a complete non-archimedean field K. Whittaker groups are defined directly via free generators of order two with fixed points in good position; the rigid spaces Fix_{P,m} and Branch_{P,m} are then constructed explicitly from this data together with theta-function parametrizations. The central morphism FB is shown to be a rigid étale covering by direct analysis of the relation between fixed-point configurations and branch loci, with explicit low-genus verifications (g=2,3) and re-derivations of semi-stable reduction properties drawn from independent prior literature (Gerritzen–Van Steen–Herrlich). No step reduces by construction to a fitted parameter, self-citation chain, or internal normalization that would make the etale-covering statement tautological.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 3 invented entities

The central claim rests on the existence of discontinuous subgroups generated by involutions whose fixed points satisfy the good-position condition, together with standard facts about rigid analytic spaces over complete non-archimedean fields. No numerical free parameters appear; the new objects (Whittaker group, Fix and Branch spaces) are defined rather than postulated with independent evidence.

axioms (2)
  • domain assumption K is a complete non-archimedean valued field whose residue field has characteristic different from 2
    Stated at the opening of the abstract as the ambient field for PGL(2,K) and the curves.
  • ad hoc to paper The fixed points {a_i, b_i} lie in good position
    Required to define the generators s_i of the Whittaker group and to equip configurations (P,m) with the rigid spaces Fix_{P,m} and Branch_{P,m}.
invented entities (3)
  • Whittaker group no independent evidence
    purpose: Discontinuous subgroup of PGL(2,K) freely generated by order-two elements with fixed points in good position
    Newly introduced to produce index-2 Schottky subgroups yielding hyperelliptic Mumford curves.
  • Fix_{P,m} no independent evidence
    purpose: Rigid space parametrizing fixed points in good position for a given configuration (P,m)
    Defined so that the morphism FB to branch-point space can be studied.
  • Branch_{P,m} no independent evidence
    purpose: Rigid space parametrizing branch points for the same configuration
    Defined as the target of the etale-covering morphism FB.

pith-pipeline@v0.9.0 · 5934 in / 1862 out tokens · 62604 ms · 2026-05-22T02:26:14.441303+00:00 · methodology

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Works this paper leans on

21 extracted references · 21 canonical work pages

  1. [1]

    Best, L.A

    A.J. Best, L.A. Betts, M. Bisatt, R.V. Bommel, V. Dokchitser, O. Faraggi, S. Kunzweiler, C. Maistret, A. Morgan, S. Muselli, and S. Nowell, A user's guide to the local arithmetic of hyperelliptic curves , Bull. Lond. Math. Soc. 54 (2022), 825--867

    work page 2022

  2. [2]

    Bosma, J

    W. Bosma, J. Cannon, C. Playoust , The M agma algebra system. I . T he user language , J. Symbolic Comput. 24 (1997), 235--265

    work page 1997

  3. [3]

    Dokchitser, V

    T. Dokchitser, V. Dokchitser, C. Maistret, A. Morgan, Arithmetic of hyperelliptic curves over local fields , Math. Ann. 385 (2023), 1213--1322

    work page 2023

  4. [4]

    Dyer and P

    J.L. Dyer and P. Scott, Periodic Automorphisms of Free Groups , Comm. Algebra 3 (1975), 195--201

    work page 1975

  5. [5]

    Fresnel and M

    J. Fresnel and M. van der Put, Rigid Analytic Geometry and Its Applications . Birkh\"auser, Boston, 2004

    work page 2004

  6. [6]

    Gerritzen and M

    L. Gerritzen and M. van der Put, Schottky groups and Mumford curves . Lecture Notes in Mathematics 817, Springer-Verlag, Berlin, 1980

    work page 1980

  7. [7]

    Herrlich, p-adisch diskontinuierlich einbettbare Graphen von Gruppen , Arch

    F. Herrlich, p-adisch diskontinuierlich einbettbare Graphen von Gruppen , Arch. Math. 39 (1982), 204--216

    work page 1982

  8. [8]

    Kadziela, Rigid Analytic Uniformization of Hyperelliptic Curves

    S. Kadziela, Rigid Analytic Uniformization of Hyperelliptic Curves . Dissertation, University of Illinois at Urbana Champaign, 2007

    work page 2007

  9. [9]

    Kadziela, Rigid Analytic Uniformization of Curves and the Study of Isogenies , Acta Appl

    S. Kadziela, Rigid Analytic Uniformization of Curves and the Study of Isogenies , Acta Appl. Math. 99 (2007),185--204

    work page 2007

  10. [10]

    E. Kaya, M. Masdeu and J.S. M\" u ller, Algorithms for heights on M umford curves , In preparation (2026)

    work page 2026

  11. [11]

    Kenku, The modular curve X_0(39) and rational isogeny , Math

    M.A. Kenku, The modular curve X_0(39) and rational isogeny , Math. Proc. Camb. Phil. Soc. 85 (1979), 21--23

    work page 1979

  12. [12]

    Masdeu and X

    M. Masdeu and X. Xarles, Efficient computation of non-archimedean theta functions, Math. Comp. 95 (2026), 457-–475

    work page 2026

  13. [13]

    Morrison and Q

    R. Morrison and Q. Ren, Algorithms for Mumford Curves , J. Symbolic Comput. 68 (2015), 259--284

    work page 2015

  14. [14]

    Mumford, An analytic construction of degenerating curves over complete local rings , Compositio Math

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

    work page 1972

  15. [15]

    van der Put, p -adic Whittaker groups , Groupe de travail d'analyse ultram\'etrique, tome 6 (1978-1979), exp

    M. van der Put, p -adic Whittaker groups , Groupe de travail d'analyse ultram\'etrique, tome 6 (1978-1979), exp. no. 15, 6 pages. https://www.numdam.org/item/GAU_1978-1979__6__A9_0.pdf

    work page 1978

  16. [16]

    van der Put and H.H

    M. van der Put and H.H. Voskuil, Discontinuous subgroups of PGL _2(K) , J. Algebra 271 (2004), 234--280

    work page 2004

  17. [17]

    Van Steen, Hyperelliptic curves defined by Schottky groups over a non- A rchimedean valued field

    G. Van Steen, Hyperelliptic curves defined by Schottky groups over a non- A rchimedean valued field . PhD thesis, University of Antwerpen, 1981

    work page 1981

  18. [18]

    Van Steen, Non-Archimedean Schottky groups and hyperelliptic curves , Nederl

    G. Van Steen, Non-Archimedean Schottky groups and hyperelliptic curves , Nederl. Akad. Wetensch. Indag. Math. 45 (1983), 97--109

    work page 1983

  19. [19]

    Teitelbaum, p-adic periods of genus two Mumford-Schottky curves , J

    J. Teitelbaum, p-adic periods of genus two Mumford-Schottky curves , J. reine angew. Math. 385 (1988), 117 --151

    work page 1988

  20. [20]

    Whittaker, On the connection of algebraic functions with automorphic functions , Philos

    E.T. Whittaker, On the connection of algebraic functions with automorphic functions , Philos. Trans. A Math. Phys. Eng. Sci. 192 (1899), 1--32. https://royalsocietypublishing.org/rsta/article/doi/10.1098/rsta.1899.0001/108241

    work page doi:10.1098/rsta.1899.0001/108241

  21. [21]

    Whittaker, On hyperlemniscate functions, a family of automorphic functions , J

    E.T. Whittaker, On hyperlemniscate functions, a family of automorphic functions , J. London Math. Soc. s1-4 (1929), 275--278

    work page 1929