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

Bertrand, Daniel; Philippon, Patrice

Ill. J. Math. 32, No. 1, 263-280 (1988).

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 \(G'\).

Reviewer: Daniel Bertrand (Paris)

MSC:

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

Keywords:

zero estimates of transcendence theory on a commutative algebraic group; periods; Abelian varieties; tori; Hilbert polynomials; Riemann-Roch theorem; lattices; Minkowski theorem

Citations:

Zbl 0516.10027; Zbl 0651.10023; Zbl 0691.14028