Equivariant deformation of Mumford curves and of ordinary curves in positive characteristic

Cornelissen, Gunther; Kato, Fumiharu

doi:10.1215/s0012-7094-03-11632-4

Cited by 30 publications

(63 citation statements)

References 21 publications

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“…Following [Bertin and Mézard 2000], in the case where G is cyclic of order p, Pries [2005] and Kontogeorgis [2007] have obtained lower and upper bounds for the dimension of the tangent space, with explicit computations in some special cases, in particular when G is an abelian p-group. (See also [Cornelissen and Kato 2003]). …”

Section: Introductionmentioning

confidence: 98%

Smooth curves having a large automorphismp-group in characteristicp >0

Matignon

Rocher

2008

ANT

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Let k be an algebraically closed field of characteristic p > 0 and C a connected nonsingular projective curve over k with genus g ≥ 2. This paper continues our study of big actions, that is, pairs (C, G) where G is a p-subgroup of the k-automorphism group of C such that |G|/g > 2 p/( p−1). If G 2 denotes the second ramification group of G at the unique ramification point of the cover C → C/G, we display necessary conditions on G 2 for (C, G) to be a big action, which allows us to pursue the classification of big actions.Our main source of examples comes from the construction of curves with many rational points using ray class field theory for global function fields, as initiated by J.-P. Serre and continued by Lauter and by Auer. In particular, we obtain explicit examples of big actions with G 2 abelian of large exponent.

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Section: Introductionmentioning

confidence: 98%

Smooth curves having a large automorphismp-group in characteristicp >0

Matignon

Rocher

2008

ANT

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“…4 , S 4 , A 5 } and three branch points. Cyclic branch groups with orders (2, 2, n), (2,3,3), (2,3,4), (2,3,5), respectively.…”

Section: The Finite Subgroups Of Pgl 2 (K)mentioning

confidence: 99%

“…The group Γ v 0 has three branch points corresponding to the maximal cyclic subgroups of Γ v 0 . The triples consisting of the orders of these cyclic groups are (2, 2, ), (2,3,3), (2,3,4) and (2,3,5) for the groups D , A 4 , S 4 and A 5 , respectively. The maximal cyclic subgroups of order 2 (resp.…”

Section: Mumford Groups With Three Branch Pointsmentioning

confidence: 99%

“…such that the kernel is a Schottky group. (5). For p = 3 there exists a surjective group homomorphisms ϕ :…”

Section: Proof Consider the Casementioning

confidence: 99%

“…The group A := Γ/Δ is a subgroup of Aut(X) such that X/A ∼ = P 1 K . In several papers [4][5][6][7][8]26,25] the construction and the classification of Mumford groups over a field K of characteristic p > 0 are studied. Here we continue this study.…”

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confidence: 99%

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Mumford curves and Mumford groups in positive characteristic

Voskuil

Put

2019

Journal of Algebra

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A Mumford group is a discontinuous subgroup Γ of PGL 2 (K), where K denotes a non archimedean valued field, such that the quotient by Γ is a curve of genus 0. As abstract group Γ is an amalgam of a finite tree of finite groups. For K of positive characteristic the large collection of amalgams having two or three branch points is classified. Using these data Mumford curves with a large group of automorphisms are discovered. A long combinatorial proof, involving the classification of the finite simple groups, is needed for establishing an upper bound for the order of the group of automorphisms of a Mumford curve. Orbifolds in the category of rigid spaces are introduced. For the projective line the relations with Mumford groups and singular stratified bundles are studied. This paper is a sequel to [P-V]. Part of it clarifies, corrects and extends work of G. Cornelissen, F. Kato and K. Kontogeorgis. 1 ated, discontinuous subgroup of PGL 2 (K) such that ∆ contains no elements ( = 1) of finite order and ∆ ∼ = {1}, Z. It turns out that ∆ is a free non-abelian group on g > 1 generators. Let Ω ⊂ P 1 K denote the rigid open subspace of ordinary points for ∆. Then X := Ω/∆ is an algebraic curve over K of genus g. The curves obtained in this way are called Mumford curves. Let Γ ⊂ PGL 2 (K) denote the normalizer of ∆. Then Γ/∆ acts on X and is in fact the group of the automorphisms of X. For K ⊃ Q p , the theme of automorphisms of Mumford curves is of interest for p-adic orbifolds and for p-adic hypergeometric differential equations. According to F. Herrlich [He] one has for Mumford curves X of genus g > 1 the bound |Aut(X)| ≤ 12(g − 1) if p > 5. For p = 2, 3, 5 there are p-adic "triangle groups" and the bounds are n p (g − 1) with n 2 = 48, n 3 = 24, n 5 = 30.In this paper we investigate the case that K has characteristic p > 0. The order of the automorphism group can be much larger than 12(g − 1). Using the Riemann-Hurwitz-Zeuthen formula one easily shows (see also the proof of Corollary 6.2):If g > 1 and |Aut(X)| > 12(g − 1), then X/Aut(X) ∼ = P 1 K and the morphism X → X/Aut(X) is branched above 2 or 3 points.There exist Mumford curves X = Ω/∆ with genus g > 1 and such that |Aut(X)| > 12(g − 1). Hence the normalizer Γ of ∆ ⊂ PGL(2, K) satisfies Ω/Γ ∼ = P 1 K . This leads to the definition of a Mumford group: This is a finitely generated, discontinuous subgroup Γ of PGL 2 (K) such that Ω/Γ ∼ = P 1 K , where Ω ⊂ P 1 K is the rigid open subset of the ordinary points for the group Γ. We exclude the possibilities that Γ is finite and that Γ contains a subgroup of finite index, isomorphic to Z. A point a ∈ P 1 K is called a branch point if a preimage b ∈ Ω of a has a non trivial stabilizer in Γ.On the other hand, a Mumford group Γ contains a normal subgroup ∆, which is of finite index and has no elements = 1 of finite order. Thus ∆ is a Schottky group, X := Ω/∆ is a Mumford curve. Above we have excluded the cases that the genus of X is 0 or 1. The group A := Γ/∆ is a subgroup of Aut(X) such that X/A ∼ = P 1 K .In several papers...

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On holomorphic polydifferentials in positive characteristic

Karanikolopoulos

2012

Mathematische Nachrichten

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ABSTRACT. In this paper we study the space Ω(m), of holomorphic m-(poly)differentials of a function field of a curve defined over an algebraically closed field of characteristic p > 0 when G is cyclic or elementary abelian group of order p n ; we give bases for each case when the base field is rational, introduce the Boseck invariants and give an elementary approach to the G module structure of Ω(m) in terms of Boseck invariants. The last computation is achieved without any restriction on the base field in the cyclic case, while in the elementary abelian case it is assumed that the base field is rational. An application to the computation of the tangent space of the deformation functor of curves with automorphisms is given.

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Equivariant deformation of Mumford curves and of ordinary curves in positive characteristic

Cited by 30 publications

References 21 publications

Smooth curves having a large automorphismp-group in characteristicp >0

Smooth curves having a large automorphismp-group in characteristicp >0

Mumford curves and Mumford groups in positive characteristic

On holomorphic polydifferentials in positive characteristic

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