Periodes et groupes de Mumford-Tate des 1-motifs

Abstract : In this thesis we study the structure and the degeneracies of the Mumford-Tate group of a 1-motive $M$ defined over $\CC$, $MT(M)$. This group is an algebraic $\QQ\,$-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's 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 $MT(M)$. The generalized Grothendieck's conjecture of periods ${\rm (CPG)}_K$ predicts that if $M$ is a 1-motive defined over an algebraically closed subfield $K$ of $\CC$, then $ {\rm deg.transc}_{\QQ}\, K ({\rm p\acute eriodes}(M))\geq \dim_{\QQ}MT( M_{\CC}).$ In the second part of this thesi we propose a conjecture of transcendance that we call {\it the elliptico-toric conjecture} (CET). Our main result is that (CET) is equivalent to ${\rm (CPG)}_K$ applied to 1-motives defined over $K$ of the kind $M=[ {\Bbb Z}^{r} \, {\buildrel u \over \longrightarrow} \,\prod^n_{j=1} {\cal E}_j \times {\GG}_m^s]$. (CET) implies some classical conjectures, as the Schanuel's conjecture or its elliptic analogue, but it implies new conjectures as well. All these conjectures following from (CET) are equivalent to ${\rm (CPG)}_K$ applied to well chosed 1-motives: for example the Schanuel's conjecture is equivalent to ${\rm (CPG)}_K$ applied to 1-motives of the kind $M=[ {\Bbb Z}^{r} \, {\buildrel u \over \longrightarrow} \, {\GG}_m^s]$.
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Theses
Mathematics [math]. Université Pierre et Marie Curie - Paris VI, 2000. French


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Contributor : Cristiana Bertolin <>
Submitted on : Friday, March 15, 2002 - 5:04:36 PM
Last modification on : Friday, March 15, 2002 - 5:04:36 PM

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Cristiana Bertolin. Periodes et groupes de Mumford-Tate des 1-motifs. Mathematics [math]. Université Pierre et Marie Curie - Paris VI, 2000. French. <tel-00001222>

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