Picard 1-motives and Tate sequences for function fields
Cornelius Greither1; Cristian Popescu2
1 Institut für Theoretische Informatik und Mathematik Universität der Bundeswehr, München 85577 Neubiberg, Germany
2 Department of Mathematics, University of California San Diego, La Jolla, CA 92093-0112, USA
Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 3.1, pp. 835-852.

We use our previous work [4] on the Galois module structure of ℓ–adic realizations of Picard 1–motives to construct explicit representatives in the ℓ–adified Tate class (i.e. explicit ℓ–adic Tate sequences, as defined in [8]) for general Galois extensions of characteristic p>0 global fields. If combined with the Equivariant Main Conjecture proved in [4], these results lead to a very direct proof of the Equivariant Tamagawa Number Conjecture for characteristic p>0 Artin motives with abelian coefficients.

En utilisant nos travaux antérieurs sur la structure des réalisations ℓ-adiques des 1-motifs de Picard en tant que modules galoisiens, nous construisons des représentants explicites pour la classe de Tate ℓ-adifiée. C’est-à-dire qu’on trouve des suites de Tate explicites, comme définies dans [8], pour une extension galoisienne générale de corps globaux en caractéristique p>0. En combinaison avec la Conjecture Principale Équivariante démontrée dans [4], ceci nous amène à une preuve assez directe de la Conjecture Équivariante des nombres de Tamagawa pour les motifs d’Artin à coefficients abéliens en caractéristique positive.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1180
Classification: 11M38, 11G20, 11G25, 11G45, 14F30
Keywords: Picard $1$–motives; étale, crystalline, and Weil-étale cohomology; Galois module structure; Tate sequences
Cornelius Greither 1; Cristian Popescu 2

1 Institut für Theoretische Informatik und Mathematik Universität der Bundeswehr, München 85577 Neubiberg, Germany
2 Department of Mathematics, University of California San Diego, La Jolla, CA 92093-0112, USA
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Cornelius Greither; Cristian Popescu. Picard 1-motives and Tate sequences for function fields. Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 3.1, pp. 835-852. doi : 10.5802/jtnb.1180. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1180/

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