On the symplectic type of isomorphisms of the p-torsion of elliptic curves

On the symplectic type of isomorphisms of the p-torsion of elliptic curves

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Détails bibliographiques
Auteurs principaux : Freitas Nuno (Auteur), Kraus Alain (Auteur)
Format : Livre
Langue : anglais
Titre complet : On the symplectic type of isomorphisms of the p-torsion of elliptic curves / Nuno Freitas, Alain Kraus
Publié : Providence (R.I.) : American Mathematical Society , 2022
Description matérielle : 1 vol. (V-105 p.)
Collection : Memoirs of the American Mathematical Society ; 1361
Sujets :
Isomorphismes (mathématiques)
Courbes elliptiques
  • Chapter 1. Motivation and results
  • 1.1 Introduction
  • 1.2 A double motivation
  • 1.3 Our approach to the problem of determining the symplectic type
  • 1.4 A complete list of local symplectic criteria at l<>p
  • Chapter 2. The existence of local symplectic criteria
  • 2.1 Existence of symplectic criteria in terms of the image of PE,p
  • 2.2 Symplectic criteria with PE,p (GQe) abelian
  • Chapter 3. The criterion in the case of good reduction
  • 3.1 The action of Frobenius
  • 3.2 Proof of Theorem 1.18
  • 3.3 A more general theorem
  • Chapter 4. Elliptic curves with potentially good reduction
  • 4.1 An useful Weierstrass model
  • 4.2 The field of good reduction
  • 4.3 The Galois group of the p-torsion field in the cases e=3, 4
  • 4.4 Proof of Theorem 1.4
  • 4.5 Proof of Lemmas 1.7, 1.8, 1.16 and 1.17
  • 4.6 The completeness of Table 1.1
  • Chapter 5. The morphism YE
  • 5.1 Explicit description of YE
  • 5.2 The morphism of YE in the tame case e=3
  • 5.3 The morphism YE in the wild case e=3
  • 5.4 The morphism YE in the tame case e=4
  • 5.5 The morphism YE in the wild case e=4
  • 5.6 The morphism YE in the wild case e=8
  • 5.7 The morphism in the wild case e=12
  • 5.8 Tables with coordinate changes
  • Chapter 6. Proof of the criteria
  • 6.1 Proof of Theorem 1.3
  • 6.2 Proof of Theorem 1.5
  • 6.3 Proof of Theorem 1.6
  • 6.4 Proof of Theorem 1.9
  • 6.5 proof of Theorem 1.10
  • 6.6 Proof of Theorem 1.22
  • 6.7 Proof of Theorems 1.13 and 1.15
  • Chapter 7. Applications
  • 7.1 Revisiting a question of Mazur
  • 7.2 The Generalized Fermat equation x2 + y3 = zp
  • 7.3 On the hyperelliptic curves y2 = xp - l and y2 = xp - 2l
  • Bibliography