Group schemes of prime order

Tate, John; Oort, Frans

doi:10.24033/asens.1186

Cited by 175 publications

(119 citation statements)

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“…We do not need this and leave the generalization to arbitrary F to the reader. Part (a) is the statement of the last corollary of [20].…”

Section: Proposition 23 Let R Be a Noetherian Ring And Letmentioning

confidence: 99%

“…Since Aut(G) has order p − 1, this implies that the Galois group acts on the points of G through a power ω i of the Teichmüller character ω. Let p be a prime of F dividing p. Since p is unramified, it follows from [20,p.15,Remark 5] that over the completion at p, the group scheme G is isomorphic to Z/pZ, µ p or an unramified twist of these group schemes. Therefore either ω i or ω 1−i is unramified at p. The character ω has order p − 1 and since p = 2, it is non-trivial and hence ramified at p. It follows that i = 0 or 1.…”

Section: Proposition 23 Let R Be a Noetherian Ring And Letmentioning

confidence: 99%

“…Then G and H aré etale over R[ 1 p ] and we can identify them with their Galois modules. The Galois action is unramified outside p. The ring R is a finite product of finite extensions of Z p over which we can apply the theory of Oort-Tate [20], Raynaud [15] and Fontaine [8,9]. Finally, the ring R[ 1 p ] ∼ = F ⊗ Q p is a product of p-adic fields.…”

Section: Proposition 23 Let R Be a Noetherian Ring And Letmentioning

confidence: 99%

“…Moreover, under assumption of the Generalized Riemann Hypothesis, every abelian variety over Q(ζ 20 ) with good reduction everywhere is isogenous to E g for some g ≥ 0.…”

Section: Theorem the Elliptic Curve E Given Bymentioning

confidence: 99%

“…Then we employ the classification theorem of Oort and Tate [20] and check that the only group schemes of order p over Z[ζ f ] are actually isomorphic to the constant group scheme Z/pZ or its Cartier dual µ p . This implies that every p-group scheme admits a filtration with closed flat subgroup schemes and subquotients isomorphic to either Z/pZ or µ p .…”

Section: ψ(X)mentioning

confidence: 99%

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Abelian varieties over cyclotomic fields with good reduction everywhere

Schoof¹

2003

Mathematische Annalen

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Abstract. For every conductor f ∈ {1, 3, 4, 5, 7, 8, 9, 11, 12, 15} there exist non-zero abelian varieties over the cyclotomic field Q(ζ f ) with good reduction everywhere. Suitable isogeny factors of the Jacobian variety of the modular curve X 1 (f ) are examples of such abelian varieties. In the other direction we show that for all f in the above set there do not exist any non-zero abelian varieties over Q(ζ f ) with good reduction everywhere except possibly when f = 11 or 15. Assuming the Generalized Riemann Hypothesis (GRH) we prove the same result when f = 11 and 15.

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“…We do not need this and leave the generalization to arbitrary F to the reader. Part (a) is the statement of the last corollary of [20].…”

Section: Proposition 23 Let R Be a Noetherian Ring And Letmentioning

confidence: 99%

Section: Proposition 23 Let R Be a Noetherian Ring And Letmentioning

confidence: 99%

Section: Proposition 23 Let R Be a Noetherian Ring And Letmentioning

confidence: 99%

“…Moreover, under assumption of the Generalized Riemann Hypothesis, every abelian variety over Q(ζ 20 ) with good reduction everywhere is isogenous to E g for some g ≥ 0.…”

Section: Theorem the Elliptic Curve E Given Bymentioning

confidence: 99%

Section: ψ(X)mentioning

confidence: 99%

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Abelian varieties over cyclotomic fields with good reduction everywhere

Schoof¹

2003

Mathematische Annalen

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Cartiermoduln lokaler Gruppenschemata der Ordnung p über einem perfekten Ring der Charakteristik p

Reimann

1988

Mathematische Nachrichten

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Bei der Behalidlung von Klassifikationsproblernen abelscher Mannigfaltigkeiten, wie sie zum Beispiel beim Beweis der MoRnELL-Vermutung oder bei der Untersuchung einiger Smummannigfdtigkeiten auftreten, ist es von groRem Nutzen, einen mijglichst guten aberblick uber die Kategorie der endlichen loknl freien Gruppenschemats uber einem fixierten Ring zii haben. Fiir den Fa11 von Gruppenschemata von Primznhlordnung haben OORT und TNm bereits 1970 eine acl hoc Klassifikation gegeben. Es erhebt sich nun die Fmge, mie sich diese Klnssifikation in modernere Methoden zur Untersuchung endlicher loknl freier Gruppeiischemnta einordnet . Eine dieser Methoden ist die Theorie der DIEuDoNKEmoduln. Nach ZITK [2] kann man jedem endlichen loknlen, lokal freien Gruppenschema uber einem perfekten Ring h' der Charakteristik p 5 0 seinen CARTIERmOdUl zuordnen. D~Z U stellt man die formale Komplettierung des gegebenen Gruppenschemas als Kern einer Isogenie von formalen Gruppen dar und betrachtet den Kokern des von der Isogenie induzierten Homomorphismus von CARTIERmOdUln. Man erhalt auf diese Weise eine Bquivalenz zmischen der Kategorie der endlichen lokalen, lokal freien Gruppenschemats uber I< und der Kategorie der DIEUDONNEmodUh endlicher Liinge uber K , auf denen V nilpotent operiert. Fur eine ausfuhrliche Darstellung verweisen wir auf [2].Andererseits hat man nnch OORT und TATE [I] eine Aquivalenz zwischen der Kategorie der endlichen lokal freien Gruppenschemata G der Ordnung p und der Kategorie der Tripe1 (L, a , b ) , wobei L ein endlich erzeugter projektiver Modul vom Rang 1 uber R, u~L @ ( p -l ) und b~L @ (~-p ) mit ab=O sind. Dabei ist fur jede kommutative K-Algebra A mit Einselement

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Tame Realisable Classes over Hopf Orders

Byott

1998

Journal of Algebra

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Let O be the ring of algebraic integers in a number field K and let G be a finite abelian group. McCulloh has characterised the classes in the locally free classgroup Ž . Cl OG which are realisable by rings of integers of tame normal extensions of K with Galois group G. We extend this to certain extensions which in general need be neither tame nor normal by replacing OG with a Hopf order ᑛ and introducing the strongly ᑛ-tame orders as an analogue of tame rings of integers. We also describe the classes realised by principal homogeneous spaces over the dual of ᑛ.

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Group schemes of prime order

Cited by 175 publications

References 2 publications

Abelian varieties over cyclotomic fields with good reduction everywhere

Abelian varieties over cyclotomic fields with good reduction everywhere

Cartiermoduln lokaler Gruppenschemata der Ordnung p über einem perfekten Ring der Charakteristik p

Tame Realisable Classes over Hopf Orders

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