Arithmetic Duality Theorems for 1-Motives

Harari, David; Szamuely, Tamás

doi:10.1515/crll.2005.2005.578.93

Cited by 35 publications

(106 citation statements)

References 15 publications

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“…Proof. By the previous corollary and the existence of the perfect "CasselsTate pairing" X 1 (A) × X 1 (B) → Q/Z ( [11], Corollary 4.9, and [8], Corollary 6.7), the dual of the second exact sequence appearing in Proposition 3.4 for B is an exact sequence…”

Section: Consequently By Lemma 44 We May Writementioning

confidence: 85%

On Néron class groups of abelian varieties

González-Avilés

2012

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. Let F be a global field, let S∞ be the set of archimedean primes of F and let S be any nonempty finite set of primes of F containing S∞. In this paper we study the Néron S-class group CA,F,S of an abelian variety A defined over F . In the well-known analogy that exists between the Birch and Swinnerton-Dyer conjecture for A over F and the analytic class number formula for the field F (in the number field case), the finite group CA,F,S ∞ (not the Tate-Shafarevich group of A) is a natural analog of the ideal class group of F .

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Section: Consequently By Lemma 44 We May Writementioning

confidence: 85%

On Néron class groups of abelian varieties

González-Avilés

2012

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…It was already shown in [Harari and Szamuely 2005] that X 0 (k, M) is finite. The finiteness results stated in the second part of the theorem are new and are also the essential part of the theorem.…”

Section: Homogeneous Self-dual Projective Bundlesmentioning

confidence: 89%

“…The idea to unify and generalise these arithmetic duality theorems to duality theorems for 1-motives appeared first in [Harari and Szamuely 2005]. Deligne constructed for each 1-motive M a dual 1-motive M ∨ .…”

Section: Homogeneous Self-dual Projective Bundlesmentioning

confidence: 99%

“…They also construct a pairing between a certain modification of X 0 (k, M) and X 2 (k, M ∨ ) and show that it is nondegenerate modulo divisible subgroups. However, this modified X 0 (k, M) remains somehow uncontrollable, and the resulting generalised pairing does not seem to be very useful (the statement of [Harari and Szamuely 2005 …”

Section: Homogeneous Self-dual Projective Bundlesmentioning

confidence: 99%

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2013

Algebra Number Theory

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This paper contains a complete proof of Fukaya and Kato's -isomorphism conjecture for invertible -modules (the case of V = V 0 (r ), where V 0 is unramified of dimension 1). Our results rely heavily on Kato's proof, in an unpublished set of lecture notes, of (commutative) -isomorphisms for one-dimensional representations of G ‫ޑ‬ p , but apart from fixing some sign ambiguities in Kato's notes, we use the theory of (φ, )-modules instead of syntomic cohomology. Also, for the convenience of the reader we give a slight modification or rather reformulation of it in the language of Fukuya and Kato and extend it to the (slightly noncommutative) semiglobal setting. Finally we discuss some direct applications concerning the Iwasawa theory of CM elliptic curves, in particular the local Iwasawa Main Conjecture for CM elliptic curves E over the extension of ‫ޑ‬ p which trivialises the p-power division points E( p) of E. In this sense the paper is complimentary to our work with Bouganis (Asian J. Math. 14:3 (2010), 385-416) on noncommutative Main Conjectures for CM elliptic curves.

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“…Il est facile de voir directementà partir de la proposition 1 que le corollaire 1 reste valable en remplaçant la complétion ∧ par la "complétion partielle" ∧ = lim ← −n ./n. On obtient alors l'analogue (d'ailleurs plus simpleà démontrer) de la proposition 2 avec cette complétion partielle grâceà la suite exacte (17) de [Harari et Szamuely 2005]. Cela suffit pour prouver le théorème 1 par la même méthode, tout enévitant les complications topologiques.…”

Section: The Shimura Liftunclassified

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2008

ANT

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Arithmetic Duality Theorems for 1-Motives

Cited by 35 publications

References 15 publications

On Néron class groups of abelian varieties

On Néron class groups of abelian varieties

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