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 W_{N} of V by
N with a natural mapping φ_{N} and then, for any points Q and Q′ of W_{N},
φ_{N}^{−1}(Q) is a homogeneous space with respect to N and there exists an
everywhere defined birational transformation of φ_{N}^{−1}(Q) onto φ_{N}^{−1}(Q′).
When N = L, there exists a bijective birational mapping of W_{L} onto A and
(W_{L},φ_{L}) is an Albanese variety of V . On the other hand, when N = D
and V is complete, W_{D} is a rational variety and φ_{D}^{−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 zetafunction of V follows from the above
result.
