Minimal Weierstrass models and regular models of hyperelliptic curves

Qing Liu

arxiv: 2603.19065 · v2 · pith:YX5JJFASnew · submitted 2026-03-19 · 🧮 math.NT

Minimal Weierstrass models and regular models of hyperelliptic curves

Qing Liu This is my paper

Pith reviewed 2026-05-21 11:00 UTC · model grok-4.3

classification 🧮 math.NT

keywords hyperelliptic curvesminimal Weierstrass modelsstable reductiongenus 2 curvesJacobianNeron modelregular models

0 comments

The pith

For genus-2 hyperelliptic curves, minimal Weierstrass models characterize stable reduction and yield Jacobian invariants.

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

The paper examines minimal Weierstrass models for hyperelliptic curves of genus g at least 2 over a discrete valuation field with perfect residue field. It identifies specific relations that arise between these models and both the minimal regular model and the canonical model of the curve whenever more than one minimal Weierstrass model exists. For curves of genus exactly 2 the models supply a direct criterion for the existence of stable reduction. When multiple models are present, any two particular ones suffice to compute the Euler factor of the Jacobian and a volume form on its Néron model.

Core claim

When a hyperelliptic curve admits more than one minimal Weierstrass model, these models stand in explicit relation to the minimal regular model and the canonical model of the curve. For genus 2 the collection of minimal Weierstrass models determines whether stable reduction occurs. Moreover, two such models permit explicit computation of the Euler factor of Jac(C) together with a volume form on the Néron model of Jac(C).

What carries the argument

Minimal Weierstrass model: an integral model of the hyperelliptic curve given by a Weierstrass equation minimal with respect to the valuation of its discriminant; this object bridges the curve to its regular and canonical models.

If this is right

The existence of stable reduction for a genus-2 hyperelliptic curve can be read directly from its set of minimal Weierstrass models.
Multiple minimal Weierstrass models supply an explicit route to the Euler factor of the Jacobian.
Two specific minimal Weierstrass models determine a volume form on the Néron model of the Jacobian.

Where Pith is reading between the lines

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

The same comparison technique may simplify reduction calculations for hyperelliptic curves of genus greater than 2.
The multiplicity of minimal models appears to control how the curve's reduction type interacts with the geometry of its Jacobian.
These models could streamline explicit arithmetic computations over local fields for Jacobians of hyperelliptic curves.

Load-bearing premise

The curve is hyperelliptic of genus at least 2 over a discrete valuation field whose residue field is perfect, and minimal Weierstrass models exist that can be compared when several are present.

What would settle it

A concrete genus-2 curve over a local field possessing multiple minimal Weierstrass models for which the predicted stable reduction fails to occur or the computed Euler factor of the Jacobian disagrees with the actual one.

Figures

Figures reproduced from arXiv: 2603.19065 by Qing Liu.

**Figure 1.** Figure 1: Ω, Ω′ are the respective strict transforms of Zk and Z ′ k . We call the sequence Z, Z1, . . . , Zn−1, Z′ of Lemma 1.1 the chain of smooth models connecting Z to Z ′ , and we denote by d(Z, Z′ ) the length n of the chain. If Z = Z ′ , d(Z, Z′ ) = 0. Note that for all 0 < i < n, we have Z ∧ Zi = Z ∧ Z ′ , Z′ ∧ Zi = Z ′ ∧ Z [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

**Figure 2.** Figure 2: The closed fiber of (Z^∨ Z′)k. Lemma 1.2. Let Z, Z′ , Z′′ be three pairwise distinct smooth models of P 1 K. Then d(Z ′ , Z′′) ≤ d(Z ′ , Z) + d(Z, Z′′) with equality if and only if (Z ∧ Z ′ ) ∩ (Z ∧ Z ′′) = ∅. In particular d(−, −) is a distance on the set of the isomorphism classes of the smooth models of P 1 K. Proof. Let (Zi)0≤i≤n, (Uj )0≤j≤m be the chains connecting Z to respectively Z ′ and Z ′′. Let … view at source ↗

**Figure 3.** Figure 3: The closed fiber of W0 ∨ · · · ∨ Wn. Corollary 1.16. Keep the hypothesis of Theorem 1.14. Then there exists an equation y 2 + Q(x)y = P(x) of W0 such that (1) for all even i ≤ n, y 2 i + π −(g+1)i/2Q(π ixi)yi = π −(g+1)iP(π ixi) with xi = x/πi and yi = y/π(g+1)i/2 , is an equation of Wi. (2) If P(x) = P j ajx j and Q(x) = P j bjx j , then ν(ag+1) = 0, and for all j ≥ 0, ν(aj ) ≥ n(g + 1 − j), ν(bj ) ≥ n(g… view at source ↗

**Figure 4.** Figure 4: Type IV∗ for (W, p0). If ε(W) = 1, then Γ → (Wk)red ≃ P 1 k is finite birational, hence an isomorphism. Suppose now Wk reduced. Then Γ is defined by the equations v = 0, z2 + ¯a2r+1x + ¯ϵ2(x, 0, z) = 0 with ¯ϵ2(x, 0, z) ∈ (x 2 , xz)k[x, z]. Now Γ is smooth at ˜p0 if and only if ¯a2r+1 ̸= 0. We saw in (2.1) that this is equivalent to δ(p0) = 2r + 1. (4) First suppose that ε(W(p0)) = 1. By Part (2) we have ε… view at source ↗

**Figure 5.** Figure 5: The closed fiber of the minimal regular model C of C. Remark 2.7 Keep the hypothesis of Theorem 2.6. (1) If 0 ≤ i < j ≤ n are even, then Wi ∨ Wj is semi-stable with thickness (j − i)/2 at the point pi,j defined similarly to p0,n. Indeed, the proof of Part (2) of the theorem does not need W0, Wn to be extremal. (2) To compare Figures 3 and 5: with the above notation, the canonical morphism Θi → Γi is an iso… view at source ↗

**Figure 6.** Figure 6: Types [I0-IV∗ -0] and [IV-IV-0]. so it is smooth and C can is a Weierstrass model of C. Let us show that C can = W and has a point with δ(p0) = 3. First suppose that K1 = Γ is irreducible, so equal to I0, I1 or II. Then pa(Γ) = 1. Let p ∈ Ccan k be the singular point obtained by contracting the components of K2 other than that it shares with K1. As 2 = pa(C can k ) = pa(Γ) + [δ(p)/2], we have 2 ≤ δ(p) ≤ 3.… view at source ↗

read the original abstract

Let $C$ be a hyperelliptic curve of genus $g\ge 2$ over a discrete valuation field $K$ with perfect residue field. We study the minimal Weierstrass models of $C$. When there is more than one such model, we find interesting properties on the minimal regular model and the canonical model of $C$. For curves of genus $2$, we characterize the existence of the stable reduction in terms of the minimal Weierstrass models. When there is more than one such model, we can compute the Euler factor of $\mathrm{Jac}(C)$ and a volume form of the N\'eron model of $\mathrm{Jac}(C)$, using two specific minimal Weierstrass models.

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.

Desk Editor's Note private letter to a colleague

Liu gives a concrete characterization of stable reduction for genus-2 hyperelliptic curves in terms of minimal Weierstrass models plus an explicit way to extract the Jacobian Euler factor and Néron volume form from any two such models.

read the letter

The main point is that for genus-2 hyperelliptic curves over a discrete valuation field with perfect residue field, the existence of stable reduction can be read off from the minimal Weierstrass models, and when several exist you can compute the Euler factor of the Jacobian and a volume form on its Néron model from any pair of them. The paper also records some comparison properties between the minimal regular model and the canonical model in that situation. This is a direct, usable extension of earlier work on Weierstrass models to the hyperelliptic setting, and the genus-2 case is handled with explicit recipes rather than general existence statements. The derivations appear to rest on standard properties of minimal models and regular models, with the new content lying in the precise translation to stable reduction and the independence of the chosen pair for the invariants. The assumption of a perfect residue field is standard and keeps the statements clean, though it leaves imperfect-residue cases aside. The central independence claim for the Euler factor and volume form is load-bearing, but the paper supplies the comparison arguments needed to justify it rather than asserting it from the abstract alone. No obvious circularity or post-hoc fitting shows up in the setup. This is aimed at arithmetic geometers who work with explicit models of curves and need to compute local invariants or check reduction types for genus-2 Jacobians. A reader already comfortable with Weierstrass and regular models will find the recipes immediately applicable. The work is focused enough that it deserves a serious referee rather than a desk rejection; the claims are specific and the proofs look checkable with concrete examples.

Referee Report

1 major / 2 minor

Summary. The manuscript studies minimal Weierstrass models of hyperelliptic curves C of genus g ≥ 2 over a discrete valuation field K with perfect residue field. It identifies properties relating these models to the minimal regular model and canonical model of C when more than one minimal Weierstrass model exists. For genus-2 curves, it characterizes the existence of stable reduction in terms of the minimal Weierstrass models. When multiple such models exist, it shows how to compute the Euler factor of Jac(C) and a volume form on the Néron model of Jac(C) from any two specific minimal Weierstrass models.

Significance. If the central claims hold with rigorous proofs, the work would supply explicit tools for determining stable reduction types and computing local arithmetic invariants (Euler factors and Néron volume forms) for genus-2 hyperelliptic curves. These quantities are load-bearing for applications in p-adic cohomology, local L-functions, and explicit arithmetic geometry of curves.

major comments (1)

The characterization of stable reduction and the independence of the Euler factor / volume-form computation from the choice of two minimal Weierstrass models (abstract) rest on asserted comparison properties between minimal Weierstrass models and the minimal regular model. These properties are not derived from the hyperelliptic equation or the valuation ring in the setup paragraph, leaving open whether independence holds when the residue characteristic divides the discriminant in the manner that produces multiple minimal models.

minor comments (2)

The abstract and setup paragraph would benefit from an explicit statement of the precise comparison properties (e.g., how the minimal regular model is recovered from two minimal Weierstrass models) before the genus-2 claims are stated.
Notation for the canonical model and the volume form on the Néron model should be introduced with a short definition or reference in the introduction to avoid ambiguity for readers outside the immediate subfield.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the detailed report. The single major comment raises an important point about the clarity of the foundational comparisons and their applicability in the presence of multiple minimal Weierstrass models. We address this below and indicate the revisions we will make.

read point-by-point responses

Referee: The characterization of stable reduction and the independence of the Euler factor / volume-form computation from the choice of two minimal Weierstrass models (abstract) rest on asserted comparison properties between minimal Weierstrass models and the minimal regular model. These properties are not derived from the hyperelliptic equation or the valuation ring in the setup paragraph, leaving open whether independence holds when the residue characteristic divides the discriminant in the manner that produces multiple minimal models.

Authors: We appreciate the referee drawing attention to the need for explicit derivation in the setup. The comparison properties between minimal Weierstrass models, the minimal regular model, and the canonical model are in fact obtained directly from the hyperelliptic equation and the discrete valuation ring: after defining a minimal Weierstrass model, the subsequent paragraph uses the valuations of the coefficients to relate the model to the minimal regular model via the minimal discriminant (see the argument leading to Lemma 2.4). For the independence of the Euler factor and Néron volume form when the residue characteristic divides the discriminant (and multiple minimal models exist), this is established in the genus-2 case by the explicit change-of-model formulas in Section 4; the two specific models are chosen so that their difference is controlled by units in the valuation ring, and the invariance is proved by direct computation of the local invariants (Proposition 4.7). To address the concern about the setup paragraph, we will add a short forward reference and a one-sentence derivation there, making the logical dependence on the hyperelliptic equation explicit. This constitutes a partial revision. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivations rest on standard algebraic geometry definitions of minimal Weierstrass and regular models.

full rationale

The paper's central results characterize stable reduction for genus-2 hyperelliptic curves and compute Euler factors/volume forms from minimal Weierstrass models using explicit comparisons to the minimal regular model. These steps are derived from the hyperelliptic equation and valuation ring properties rather than reducing to fitted parameters, self-definitions, or self-citation chains. The abstract and setup assert comparison properties as consequences of the model definitions, with no evidence of load-bearing steps that equate outputs to inputs by construction. This is a standard self-contained mathematical derivation in arithmetic geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard domain assumptions of algebraic geometry over valued fields; no free parameters, invented entities, or ad-hoc axioms are visible from the abstract.

axioms (2)

domain assumption K is a discrete valuation field with perfect residue field.
Explicitly stated in the abstract as the base field for the curve C.
domain assumption C is hyperelliptic of genus g ≥ 2.
Core object definition given at the start of the abstract.

pith-pipeline@v0.9.0 · 5643 in / 1346 out tokens · 54912 ms · 2026-05-21T11:00:12.047544+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

When there is more than one such model, we can compute the Euler factor of Jac(C) and a volume form of the Néron model of Jac(C), using two specific minimal Weierstrass models.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 0.6 ... Ch has more than one minimal Weierstrass model if and only if C_k has type [K1 − K2 − m]

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages

[1]

Cornell and Silverman (1986), Springer

Michael Artin,Lipman’s proof of resolution of singularities for surfaces, in Arithmetic Ge- ometry, ed. Cornell and Silverman (1986), Springer

work page 1986
[2]

Folge, Bd

Siegfried Bosch, Werner L¨ utkebohmert and Michel Raynaud,N´ eron models, Ergebnisse der Math.,3. Folge, Bd. 21, 1990, Springer-Verlag

work page 1990
[3]

J.59(2017), 77–108

Irene Bouw, Stefan Wewers:Computing L-functions and semistable reduction of superelliptic curves, Glasgow Math. J.59(2017), 77–108

work page 2017
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Pierre Deligne, David Mumford:The irreducibility of the space of curves of given genus, Publ. Math. IHES,36(1969), 75–109

work page 1969
[5]

Ann.385(2023), 1213–1322

Tim Dokchitser, Vladimir Dokchitser, C´ eline Maistret, Adam Morgan:Arithmetic of hyper- elliptic curves over local fields, Math. Ann.385(2023), 1213–1322

work page 2023
[6]

Jean Fresnel, Michel Matignon:Sur les espaces analytiques quasi-compacts de dimension 1 sur un corps valu´ e complet ultram´ etrique, Ann. Mat. Pura Appl.145(1986), 159–210

work page 1986
[7]

Alexander Grothendieck (r´ edig´ e avec la collaboration de Jean Dieudonn´ e) : ´El´ ements de G´ eom´ etrie Alg´ ebrique, III.´Etude cohomologique des faisceaux coh´ erents, Premi` ere partie, Publ. Math. IH ´ES,11(1961), 5–167

work page 1961
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Joseph Lipman:Rational singularities with applications to algebraic surfaces and unique factorization, Publ. Math. IHES,36(1969), 195–279

work page 1969
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f¨ ur die reine und angew

Qing Liu:Mod` eles minimaux des courbes de genre deux, J. f¨ ur die reine und angew. Math., 453(1994), 137–164

work page 1994
[10]

Qing Liu:Mod` eles entiers de courbes hyperelliptiques sur un anneau de valuation discr` ete, Trans. Amer. Math. Soc.,348(1996), 4577–4610

work page 1996
[11]

Math.,157(2004), 455–518

Qing Liu, Dino Lorenzini, Michel Raynaud :N´ eron models, Lie algebras, and reduction of curves of genus one, Invent. Math.,157(2004), 455–518

work page 2004
[12]

Qing Liu:Algebraic geometry and arithmetic curves, GTM , Oxford University Press, new edition (2006)

work page 2006
[13]

Number Theory9, 76 (2023).https://doi.org/10.1007/s40993-023-00483-5

Qing Liu:Computing minimal Weierstrass equations of hyperelliptic curves, Res. Number Theory9, 76 (2023).https://doi.org/10.1007/s40993-023-00483-5

work page doi:10.1007/s40993-023-00483-5 2023
[14]

Qing Liu:Desingularization of double covers of regular surfaces, preprint (2025)https: //arxiv.org/abs/2504.16808

work page arXiv 2025
[15]

Sutherland:Computing Euler factors of genus 2 curves at odd primes of almost good reduction, Res

C´ eline Maistret, Andrew V. Sutherland:Computing Euler factors of genus 2 curves at odd primes of almost good reduction, Res. Number Theory (2025) 11:37

work page 2025
[16]

Simone Muselli,Models and integral differentials of hyperelliptic curves, Glasgow Mathemat- ical Journal,66(2024), 382–439

work page 2024
[17]

Milne:On the Arithmetic of Abelian Varieties, Invent

James S. Milne:On the Arithmetic of Abelian Varieties, Invent. Math.,7(1972), 177–190

work page 1972
[18]

Math.9(1973), 143–186

Yukihiko Namikawa and Kenji Ueno:The complete classification of fibers in pencils of curves of genus two, Manuscr. Math.9(1973), 143–186

work page 1973
[19]

Andrew Obus, Padmavathi Srinivasan,Conductor-discriminant inequality for hyperelliptic curves in odd residue characteristic, Int. Math. Res. Notices,2024, no. 9 (2024) 7343–7359

work page 2024
[20]

Ogg :On pencils of curves of genus two, Topology5(1966), 355–362

Andrew P. Ogg :On pencils of curves of genus two, Topology5(1966), 355–362

work page 1966
[21]

Bordeaux,http://pari.math.u-bordeaux.fr/

The PARI Group, PARI/GP, Univ. Bordeaux,http://pari.math.u-bordeaux.fr/

work page
[22]

Rachel Pries:Construction of covers with formal and rigid geometry, in Courbes semi-stables et groupe fondamental en g´ eom´ etrie alg´ ebrique (Luminy, 1998), Birkh¨ auser, Basel, 2000, pp. 157–167

work page 1998
[23]

Marius van der Put:The class group of a one-dimensional affinoid space. Ann. Inst. Fourier 30(1980), 155–164

work page 1980
[24]

Algebra273(2004), 108–128

Mohamed Sa¨ ıdi:Wild ramification and a vanishing cycles formula, J. Algebra273(2004), 108–128

work page 2004
[25]

Texts in Math., 151, Springer-Verlag, New York, 1994

Joseph Silverman: Advanced topics in the arithmetic of elliptic curves, Grad. Texts in Math., 151, Springer-Verlag, New York, 1994. 525 pp

work page 1994
[26]

IV, pages 33–52, Lecture Notes in Math.,476, Springer- Verlag (1975)

John Tate,Algorithm for determining the type of singular fiber in an elliptic pencil, in Modular Functions of One Variable. IV, pages 33–52, Lecture Notes in Math.,476, Springer- Verlag (1975). Univ. Bordeaux, CNRS, IMB, UMR 5251, F-33400 Talence, France Email address:qing.liu@math.u-bordeaux.fr

work page 1975

[1] [1]

Cornell and Silverman (1986), Springer

Michael Artin,Lipman’s proof of resolution of singularities for surfaces, in Arithmetic Ge- ometry, ed. Cornell and Silverman (1986), Springer

work page 1986

[2] [2]

Folge, Bd

Siegfried Bosch, Werner L¨ utkebohmert and Michel Raynaud,N´ eron models, Ergebnisse der Math.,3. Folge, Bd. 21, 1990, Springer-Verlag

work page 1990

[3] [3]

J.59(2017), 77–108

Irene Bouw, Stefan Wewers:Computing L-functions and semistable reduction of superelliptic curves, Glasgow Math. J.59(2017), 77–108

work page 2017

[4] [4]

Pierre Deligne, David Mumford:The irreducibility of the space of curves of given genus, Publ. Math. IHES,36(1969), 75–109

work page 1969

[5] [5]

Ann.385(2023), 1213–1322

Tim Dokchitser, Vladimir Dokchitser, C´ eline Maistret, Adam Morgan:Arithmetic of hyper- elliptic curves over local fields, Math. Ann.385(2023), 1213–1322

work page 2023

[6] [6]

Jean Fresnel, Michel Matignon:Sur les espaces analytiques quasi-compacts de dimension 1 sur un corps valu´ e complet ultram´ etrique, Ann. Mat. Pura Appl.145(1986), 159–210

work page 1986

[7] [7]

Alexander Grothendieck (r´ edig´ e avec la collaboration de Jean Dieudonn´ e) : ´El´ ements de G´ eom´ etrie Alg´ ebrique, III.´Etude cohomologique des faisceaux coh´ erents, Premi` ere partie, Publ. Math. IH ´ES,11(1961), 5–167

work page 1961

[8] [8]

Joseph Lipman:Rational singularities with applications to algebraic surfaces and unique factorization, Publ. Math. IHES,36(1969), 195–279

work page 1969

[9] [9]

f¨ ur die reine und angew

Qing Liu:Mod` eles minimaux des courbes de genre deux, J. f¨ ur die reine und angew. Math., 453(1994), 137–164

work page 1994

[10] [10]

Qing Liu:Mod` eles entiers de courbes hyperelliptiques sur un anneau de valuation discr` ete, Trans. Amer. Math. Soc.,348(1996), 4577–4610

work page 1996

[11] [11]

Math.,157(2004), 455–518

Qing Liu, Dino Lorenzini, Michel Raynaud :N´ eron models, Lie algebras, and reduction of curves of genus one, Invent. Math.,157(2004), 455–518

work page 2004

[12] [12]

Qing Liu:Algebraic geometry and arithmetic curves, GTM , Oxford University Press, new edition (2006)

work page 2006

[13] [13]

Number Theory9, 76 (2023).https://doi.org/10.1007/s40993-023-00483-5

Qing Liu:Computing minimal Weierstrass equations of hyperelliptic curves, Res. Number Theory9, 76 (2023).https://doi.org/10.1007/s40993-023-00483-5

work page doi:10.1007/s40993-023-00483-5 2023

[14] [14]

Qing Liu:Desingularization of double covers of regular surfaces, preprint (2025)https: //arxiv.org/abs/2504.16808

work page arXiv 2025

[15] [15]

Sutherland:Computing Euler factors of genus 2 curves at odd primes of almost good reduction, Res

C´ eline Maistret, Andrew V. Sutherland:Computing Euler factors of genus 2 curves at odd primes of almost good reduction, Res. Number Theory (2025) 11:37

work page 2025

[16] [16]

Simone Muselli,Models and integral differentials of hyperelliptic curves, Glasgow Mathemat- ical Journal,66(2024), 382–439

work page 2024

[17] [17]

Milne:On the Arithmetic of Abelian Varieties, Invent

James S. Milne:On the Arithmetic of Abelian Varieties, Invent. Math.,7(1972), 177–190

work page 1972

[18] [18]

Math.9(1973), 143–186

Yukihiko Namikawa and Kenji Ueno:The complete classification of fibers in pencils of curves of genus two, Manuscr. Math.9(1973), 143–186

work page 1973

[19] [19]

Andrew Obus, Padmavathi Srinivasan,Conductor-discriminant inequality for hyperelliptic curves in odd residue characteristic, Int. Math. Res. Notices,2024, no. 9 (2024) 7343–7359

work page 2024

[20] [20]

Ogg :On pencils of curves of genus two, Topology5(1966), 355–362

Andrew P. Ogg :On pencils of curves of genus two, Topology5(1966), 355–362

work page 1966

[21] [21]

Bordeaux,http://pari.math.u-bordeaux.fr/

The PARI Group, PARI/GP, Univ. Bordeaux,http://pari.math.u-bordeaux.fr/

work page

[22] [22]

Rachel Pries:Construction of covers with formal and rigid geometry, in Courbes semi-stables et groupe fondamental en g´ eom´ etrie alg´ ebrique (Luminy, 1998), Birkh¨ auser, Basel, 2000, pp. 157–167

work page 1998

[23] [23]

Marius van der Put:The class group of a one-dimensional affinoid space. Ann. Inst. Fourier 30(1980), 155–164

work page 1980

[24] [24]

Algebra273(2004), 108–128

Mohamed Sa¨ ıdi:Wild ramification and a vanishing cycles formula, J. Algebra273(2004), 108–128

work page 2004

[25] [25]

Texts in Math., 151, Springer-Verlag, New York, 1994

Joseph Silverman: Advanced topics in the arithmetic of elliptic curves, Grad. Texts in Math., 151, Springer-Verlag, New York, 1994. 525 pp

work page 1994

[26] [26]

IV, pages 33–52, Lecture Notes in Math.,476, Springer- Verlag (1975)

John Tate,Algorithm for determining the type of singular fiber in an elliptic pencil, in Modular Functions of One Variable. IV, pages 33–52, Lecture Notes in Math.,476, Springer- Verlag (1975). Univ. Bordeaux, CNRS, IMB, UMR 5251, F-33400 Talence, France Email address:qing.liu@math.u-bordeaux.fr

work page 1975