## Algebraic subgroups of commutative algebraic groups. (Sous-groupes algébriques de groupes algébriques commutatifs.)(French)Zbl 0618.14020

When applied with “effective” purposes in mind, the zero estimates of transcendence theory on a commutative algebraic group $$G$$ defined over $$\mathbb C$$ require a precise description of the algebraic subgroups $$G'$$ of $$G$$ whose ideal of definition has a given degree. This problem was studied by D. W. Masser and G. Wüstholz [Invent. Math. 72, 407–464 (1983; Zbl 0516.10027)] when $$G$$ is a power of an elliptic curve, and appears in a general context in the recent work of P. Philippon and M. Waldschmidt [”Formes linéaires de logarithmes sur les groupes algébriques commutatifs”, Ill. J. Math. 32, No. 2, 281–314 (1988; Zbl 0651.10023)]. The solution given here consists in bounding from below the degree of $$G'$$ by the volume of the maximal compact subgroup of $$G'(\mathbb C)$$. (These quantities are in fact shown to be equivalent when $$G$$ is the product of an abelian variety by a linear group; this result has just been extended to the general case by H. Lange [”A remark on the degrees of commutative algebraic groups”, Preprint MSRI (Berkeley 1987); Ill. J. Math. 33, No. 3, 409–415 (1989; Zbl 0691.14028)]. Classical arguments from the geometry of numbers then provide the bounds required in the applications for the periods of $$G'$$.

### MSC:

 14L10 Group varieties 14K05 Algebraic theory of abelian varieties 11J81 Transcendence (general theory) 11H06 Lattices and convex bodies (number-theoretic aspects)

### Citations:

Zbl 0516.10027; Zbl 0651.10023; Zbl 0691.14028