The Mumford-Tate group of 1-motives

Cristiana Bertolin[1]

  • [1] Université Louis Pasteur, UFR de Mathématiques et Informatique, 7 rue René Descartes, 67084 Strasbourg Cedex (France)

Annales de l’institut Fourier (2002)

  • Volume: 52, Issue: 4, page 1041-1059
  • ISSN: 0373-0956

Abstract

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In this paper we study the structure and the degeneracies of the Mumford-Tate group M T ( M ) of a 1-motive M defined over ℂ . This group is an algebraic ℚ - group acting on the Hodge realization of M and endowed with an increasing filtration W • . We prove that the unipotent radical of M T ( M ) , which is W - 1 ( M T ( M ) ) , injects into a “generalized” Heisenberg group. We then explain how to reduce to the study of the Mumford-Tate group of a direct sum of 1-motives whose torus’character group and whose lattice are both of rank 1. Next we classify and we study the degeneracies of M T ( M ) , i.e., those phenomena which imply the decrement of the dimension of M T ( M ) .

How to cite

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Bertolin, Cristiana. "The Mumford-Tate group of 1-motives." Annales de l’institut Fourier 52.4 (2002): 1041-1059. <http://eudml.org/doc/116002>.

@article{Bertolin2002,
abstract = {In this paper we study the structure and the degeneracies of the Mumford-Tate group $MT(M)$ of a 1-motive $M$ defined over $\{\mathbb \{C\}\}$. This group is an algebraic $\{\mathbb \{Q\}\}$- group acting on the Hodge realization of $M$ and endowed with an increasing filtration $W_\bullet $. We prove that the unipotent radical of $MT(M)$, which is $W_\{-1\}(MT(M))$, injects into a “generalized” Heisenberg group. We then explain how to reduce to the study of the Mumford-Tate group of a direct sum of 1-motives whose torus’character group and whose lattice are both of rank 1. Next we classify and we study the degeneracies of $MT(M)$, i.e., those phenomena which imply the decrement of the dimension of $MT(M)$.},
affiliation = {Université Louis Pasteur, UFR de Mathématiques et Informatique, 7 rue René Descartes, 67084 Strasbourg Cedex (France)},
author = {Bertolin, Cristiana},
journal = {Annales de l’institut Fourier},
keywords = {1-motives; Mumford-Tate group; degeneracies; Poincaré biextension; Heisenberg group},
language = {eng},
number = {4},
pages = {1041-1059},
publisher = {Association des Annales de l'Institut Fourier},
title = {The Mumford-Tate group of 1-motives},
url = {http://eudml.org/doc/116002},
volume = {52},
year = {2002},
}

TY - JOUR
AU - Bertolin, Cristiana
TI - The Mumford-Tate group of 1-motives
JO - Annales de l’institut Fourier
PY - 2002
PB - Association des Annales de l'Institut Fourier
VL - 52
IS - 4
SP - 1041
EP - 1059
AB - In this paper we study the structure and the degeneracies of the Mumford-Tate group $MT(M)$ of a 1-motive $M$ defined over ${\mathbb {C}}$. This group is an algebraic ${\mathbb {Q}}$- group acting on the Hodge realization of $M$ and endowed with an increasing filtration $W_\bullet $. We prove that the unipotent radical of $MT(M)$, which is $W_{-1}(MT(M))$, injects into a “generalized” Heisenberg group. We then explain how to reduce to the study of the Mumford-Tate group of a direct sum of 1-motives whose torus’character group and whose lattice are both of rank 1. Next we classify and we study the degeneracies of $MT(M)$, i.e., those phenomena which imply the decrement of the dimension of $MT(M)$.
LA - eng
KW - 1-motives; Mumford-Tate group; degeneracies; Poincaré biextension; Heisenberg group
UR - http://eudml.org/doc/116002
ER -

References

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  8. P. Deligne, Hodge cycles on abelian varieties, Hodge cycles, motives and Shimura varieties, 900 (1982), Springer L.N. Zbl0537.14006
  9. P. Deligne, Letter to the author, (2001) 
  10. P. Deligne, J.-S. Milne, Tannakian categories, Hodge cycles, motives and Shimura varieties 900 (1982), Springer Zbl0477.14004
  11. O. Jacquinot, K. Ribet, Deficient points on extensions of abelian varieties by 𝔾 m , J. Number Th. 25 (1987), 133-151 Zbl0667.14021MR873872
  12. H. Lange, C. Birkenhake, Complex abelian varieties, 302 (1992), Springer-Verlag Zbl0779.14012MR1217487
  13. N. Saavedra, Rivano, Catégories tannakiennes, 265 (1972), Springer Zbl0241.14008MR338002

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