Periods of 1-motives and transcendence

The generalized Grothendieck's conjecture of periods (CPG)(K) predicts that if M is a 1-motive defined over an algebraically closed subfield K of C, then deg.transc(Q) K(periodes(M)) greater than or equal to dim(Q) MT(M-C). In this article we propose a conjecture of transcendance that we call the elliptico-toric conjecture (CET). Our main result is that (CET) is equivalent to (CPG)K applied to 1-motives defined over K of the kind M = [Z(r) -->(u) Pi(j=1)(n) E-j x G(m)(s)]. (CET) implies some classical conjectures, as the in Schanuel's conjecture or its elliptic analogue, but it implies new conjectures as well. All these conjectures following from (CET) are equivalent to (CPG)K applied to well chosed 1-motives: for example the Schanuel's conjecture is equivalent to (CPG)K applied to I-motives of the kind M [Z(r) -->(u) G(m)(s)]. (C) 2002 Published by Elsevier Science (USA).