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 
@article{JTNB_2021__33_3.1_835_0,
     author = {Cornelius Greither and Cristian Popescu},
     title = {Picard 1-motives and {Tate} sequences for function fields},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {835--852},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {33},
     number = {3.1},
     year = {2021},
     doi = {10.5802/jtnb.1180},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1180/}
}
TY  - JOUR
AU  - Cornelius Greither
AU  - Cristian Popescu
TI  - Picard 1-motives and Tate sequences for function fields
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2021
SP  - 835
EP  - 852
VL  - 33
IS  - 3.1
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1180/
DO  - 10.5802/jtnb.1180
LA  - en
ID  - JTNB_2021__33_3.1_835_0
ER  - 
%0 Journal Article
%A Cornelius Greither
%A Cristian Popescu
%T Picard 1-motives and Tate sequences for function fields
%J Journal de théorie des nombres de Bordeaux
%D 2021
%P 835-852
%V 33
%N 3.1
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1180/
%R 10.5802/jtnb.1180
%G en
%F JTNB_2021__33_3.1_835_0
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/

[1] David Burns On the values of equivariant zeta functions of curves over finite fields, Doc. Math., Volume 9 (2004), pp. 357-399 | MR | Zbl

[2] David Burns; Matthias Flach On Galois module invariants associated to Tate motives, Am. J. Math., Volume 120 (1998) no. 6, pp. 1343-1397 | DOI | Zbl

[3] Pierre Deligne Théorie de Hodge, III, Publ. Math., Inst. Hautes Étud. Sci., Volume 44 (1974), pp. 5-77 | DOI | Numdam | Zbl

[4] Cornelius Greither; Cristian D. Popescu The Galois module structure of ℓ-adic realizations of Picard 1-motives and applications, Int. Math. Res. Not., Volume 2012 (2012) no. 5, pp. 986-1036 | DOI | MR | Zbl

[5] Cornelius Greither; Cristian D. Popescu An equivariant main conjecture in Iwasawa theory and applications, J. Algebr. Geom., Volume 24 (2015) no. 4, pp. 629-692 | DOI | MR | Zbl

[6] Cornelius Greither; Cristian D. Popescu Abstract ℓ–adic 1-motives and Tate’s canonical class for number fields, Doc. Math., Volume 23 (2018), pp. 839-870 | MR | Zbl

[7] Andreas Nickel Equivariant Iwasawa theory and non-abelian Stark-type conjectures, Proc. Lond. Math. Soc., Volume 106 (2013) no. 6, pp. 1223-1247 | DOI | MR | Zbl

[8] John Tate The cohomology groups of tori in finite Galois extensions of number fields, Nagoya Math. J., Volume 27 (1966), pp. 709-719 | DOI | MR | Zbl

[9] John Tate Les conjectures de Stark sur les fonctions L d’Artin en s=0, Progress in Mathematics, 47, Birkhäuser, 1984

[10] Malte Witte Non-commutative Iwasawa theory for global fields, 2017 (Habilitationsschrift, Heidelberg)

[11] Malte Witte Non-commutative L-functions for p-adic representations over totally real fields, Doc. Math., Volume 24 (2019), pp. 1413-1511 | MR | Zbl

Cited by Sources: