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 a′ of A, α−1(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
α−1(a) onto α−1(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 Q′ of WN,
φN−1(Q) is a homogeneous space with respect to N and there exists an
everywhere defined birational transformation of φN−1(Q) onto φN−1(Q′).
When N = L, there exists a bijective birational mapping of WL onto A and
(WL,φL) is an Albanese variety of V . On the other hand, when N = D
and V is complete, WD is a rational variety and φD−1(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.
