#### Vol. 15, No. 2, 1965

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On algebraic homogeneous spaces

### Makoto Ishida

Vol. 15 (1965), No. 2, 525–535
##### Abstract

Let V be a homogeneous space with respect to a connected algebraic group G and (A,α) an Albanese variety of V . Then, for any points a and aof A, α1(a) is a homogeneous space, of dimension = dimV dimA, with respect to the maximal connected linear normal algebraic subgroup L of G and there exists an everywhere defined birational transformation of α1(a) onto α1(a). We have (a) dimA = 0 V is considered as a homogeneous space with respect to a connected linear algebraic group The isotropy group of any point on V contains D (where D is the smallest normal algebraic subgroup of G giving rise to a linear factor group); (b) dimA = dimV V is considered as a homogeneous space with respect to an abelian variety The isotropy group of any point on V contains L. More generally, for a connected normal algebraic subgroup N of G, we can define a quotient variety WN of V by N with a natural mapping φN and then, for any points Q and Qof WN, φN1(Q) is a homogeneous space with respect to N and there exists an everywhere defined birational transformation of φN1(Q) onto φN1(Q). When N = L, there exists a bijective birational mapping of WL onto A and (WLL) is an Albanese variety of V . On the other hand, when N = D and V is complete, WD is a rational variety and φD1(Q) is birationally equivalent to the direct product of an abelian variety and a rational variety. In the case where the definition field k of the homogeneous space V is finite, there exists a homogeneous space W with respect to L, defined over k, such that we have (the number of rational points on V over k) =(the number of rational points on A over k) × (the number of rational points on W over k). In particular, if V is complete then the conjecture of Lang and Weil on the zeros of the congruence zeta-function of V follows from the above result.

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