## Chow’s K/k-image and K/k-trace, and the Lang-Néron theorem (2006)

Venue: | Enseign. Math |

Citations: | 13 - 1 self |

### BibTeX

@ARTICLE{Conrad06chow’sk/k-image,

author = {Brian Conrad},

title = {Chow’s K/k-image and K/k-trace, and the Lang-Néron theorem},

journal = {Enseign. Math},

year = {2006}

}

### OpenURL

### Abstract

Let K/k be an extension of fields, and assume that it is primary: the algebraic closure of k in K is purely inseparable over k. The most interesting case in practice is when K/k is a regular extension: K/k is separable and k is algebraically closed in K. Regularity is automatic if k is perfect. (For K/k finitely generated, regularity is equivalent to K arising as the function field of a smooth and geometrically connected

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Citation Context ...focus attention on positive-dimensional components. Any two k-points on an irreducible finite-type k-scheme V of positive dimension lie in a common irreducible curve X in V (see the Lemma on p. 56 in =-=[25]-=-), so it remains to check that if P, P ′ : C ⇒ Ã are two C-maps lying in an algebraic family of maps parameterized by an irreducible k-curve X then P and P ′ coincide in A(K)/TrK/k(A)(k). To be precis... |

253 | Geometric Invariant Theory - Mumford, Fogarty, et al. - 1994 |

240 |
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(Show Context)
Citation Context ...how ([3], [4]) and the book on abelian varieties by Lang [18] discuss the K/k-image and K/k-trace and develop their properties, but entirely in Weil’s framework [34]. Similarly, in Lang’s modern book =-=[20]-=- the Lang–Néron theorem is proved in Weil’s language. In connection with my work in [5], where the Lang-Néron theorem plays a crucial role, I was motivated to write this expository account of a scheme... |

199 |
Néron models
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Citation Context .../k(AK ′) such that the diagram TrK ′/k(AK ′)K ′ [γ] K ′ Tr K ′ /k(AK ′)K ′ i −1 γ ≃ � γ∗ (TrK ′/k(AK ′)K ′) γ ∗ (τAK′ ,K′ /k) ������ ������ � AK ′ γ∗ (AK′) ≃ τ AK ′ ,K ′ /k commutes. Uniqueness gives =-=[1]-=- = id and [γ1γ2] = [γ1] ◦ [γ2], so each [γ] is an automorphism and we get a natural action of the finite group Gal(K ′/K) on the abelian variety TrK ′/k(AK ′) over k. For x ∈ Tr K ′ /k(AK ′)(K) and y ... |

176 | Arithmetic moduli of elliptic curves - Katz, Mazur - 1985 |

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Techniques de construction et theoremes d'existence en geometrie algebrique IV. Les schemas de Hilbert, Seminaire
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Citation Context ...ixed degree (or equivalently, a fixed Hilbert polynomial) has finite image in A(K)/Tr K/k(A)(k). By the quasi-compactness aspects of Grothendieck’s representability results on Hilbert and Hom-schemes =-=[10]-=-, the functor of morphisms P : C → Ã such that pr 2 ◦ P = 1C and fP has degree δ in P (n+1)(m+1)−1 is represented by the “degree-δ” Hom-scheme Hδ that is of finite type over k. Thus, it suffices to re... |

113 | Degeneration of abelian varieties
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Citation Context ...e., [−1] ∗ (Nφ) ≃ Nφ), where P is the Poincaré bundle on A × A∨ . There are many such φ for any A, such as φ = φL : x ↦→ t∗ x(L ) ⊗ L −1 for any ample symmetric line bundle L on A, in which case Nφ = =-=[2]-=- ∗ (L ) ⊗ L ⊗(−2) . For any regular and finitely generated extension of fields K/k and any polarized abelian variety (A, φ) over K such that (1, φ) ∗ (P) is symmetric, we wish to put a similar structu... |

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Citation Context ...he language of schemes. The papers of Chow ([3], [4]) and the book on abelian varieties by Lang [18] discuss the K/k-image and K/k-trace and develop their properties, but entirely in Weil’s framework =-=[34]-=-. Similarly, in Lang’s modern book [20] the Lang–Néron theorem is proved in Weil’s language. In connection with my work in [5], where the Lang-Néron theorem plays a crucial role, I was motivated to wr... |

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Citation Context ...ection concerning K-heights, see [14, §B] and [25, §4, Appendix II], where proofs are given for number fields but carry over essentially verbatim to any generalized global field. Many basic proofs in =-=[14]-=- are written with restrictive smoothness hypotheses, though as noted in [14, B.3.6] such hypotheses can be avoided with better definitions in terms of Cartier divisors rather than Weil divisors. (The ... |

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Citation Context ...ies over finitely generated regular extensions K/k with respect to some field of “constants” k, there is a generalization of the Mordell–Weil theorem, due to Néron [26] (in his thesis) and Lang–Néron =-=[19]-=-, and in this theorem a crucial role is played by the K/k-trace and the K/k-image of an abelian variety A over K. These constructions are also ubiquitous in many problems concerning families of abelia... |

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The isogeny class of a CM-type abelian variety is defined over a finite extension of the prime field
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Citation Context ... done and hence the result really is non-trivial.) The key to constructing the right abelian variety over a finite extension of the prime field is to form a suitable K/k-trace. We refer the reader to =-=[27]-=- for an exposition of Grothendieck’s proof. In §3 of Raynaud’s Bourbaki report [28] on Grothendieck’s generalization of the Ogg–Shafarevich formula, the reader can find some additional elegant applica... |

20 |
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Citation Context ...t abelian variety over a finite extension of the prime field is to form a suitable K/k-trace. We refer the reader to [27] for an exposition of Grothendieck’s proof. In §3 of Raynaud’s Bourbaki report =-=[28]-=- on Grothendieck’s generalization of the Ogg–Shafarevich formula, the reader can find some additional elegant applications of the Lang–Néron theorem. Some more recent papers that apply the Lang–Néron ... |

18 |
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Citation Context ...me.) In the theory of abelian varieties over finitely generated regular extensions K/k with respect to some field of “constants” k, there is a generalization of the Mordell–Weil theorem, due to Néron =-=[26]-=- (in his thesis) and Lang–Néron [19], and in this theorem a crucial role is played by the K/k-trace and the K/k-image of an abelian variety A over K. These constructions are also ubiquitous in many pr... |

12 | théorèmes de finitude pour le foncteur de Picard, in Théorie des intersections et théorème de - Kleiman, Les - 1971 |

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9 |
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Citation Context ...rem and discuss constructions of the K/k-image and K/k-trace for finitely generated regular extensions K/k are [15] (which gives a construction of the K/k-image using Albanese varieties) and [13] and =-=[29]-=- (which give Raynaud’s construction of the K/k-trace using Picard varieties). Terminology and Notation. For any field k, a k-variety is a separated and geometrically integral k-scheme of finite type. ... |

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Sur le groupe des classes d’un schéma arithmétique (with an appendix by M. Hindry), preprint
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(Show Context)
Citation Context ...cations of the Lang–Néron theorem. Some more recent papers that apply the Lang–Néron theorem and discuss constructions of the K/k-image and K/k-trace for finitely generated regular extensions K/k are =-=[15]-=- (which gives a construction of the K/k-image using Albanese varieties) and [13] and [29] (which give Raynaud’s construction of the K/k-trace using Picard varieties). Terminology and Notation. For any... |

5 | Root numbers and ranks in positive characteristic
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(Show Context)
Citation Context ...K/k-trace and develop their properties, but entirely in Weil’s framework [34]. Similarly, in Lang’s modern book [20] the Lang–Néron theorem is proved in Weil’s language. In connection with my work in =-=[5]-=-, where the Lang-Néron theorem plays a crucial role, I was motivated to write this expository account of a scheme-theoretic approach to Chow’s results and the Lang–Néron theorem. In some instances the... |

5 |
Fibrations et conjecture de Tate
- Hindry, Pacheco, et al.
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(Show Context)
Citation Context ...éron theorem and discuss constructions of the K/k-image and K/k-trace for finitely generated regular extensions K/k are [15] (which gives a construction of the K/k-image using Albanese varieties) and =-=[13]-=- and [29] (which give Raynaud’s construction of the K/k-trace using Picard varieties). Terminology and Notation. For any field k, a k-variety is a separated and geometrically integral k-scheme of fini... |

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2 |
Abelian varieties, Interscience Tract 7
- Lang
- 1959
(Show Context)
Citation Context ...c geometry, unfortunately no detailed general reference on these topics has been available entirely in the language of schemes. The papers of Chow ([3], [4]) and the book on abelian varieties by Lang =-=[18]-=- discuss the K/k-image and K/k-trace and develop their properties, but entirely in Weil’s framework [34]. Similarly, in Lang’s modern book [20] the Lang–Néron theorem is proved in Weil’s language. In ... |

1 |
On abelian varieties over function fields
- Chow
- 1955
(Show Context)
Citation Context ...the K/k-trace is the largest abelian subvariety of A that can be defined over k. A precise description along these lines requires some care in positive characteristic. These concepts are due to Chow (=-=[3]-=-, [4]). Despite the importance of Chow’s K/k-trace and K/k-image and the Lang–Néron theorem in arithmetic geometry, unfortunately no detailed general reference on these topics has been available entir... |