RÉSUMÉ

Let $G$ be an affine algebraic group , $X$ a right $G$-homogeneous space both defined over an algebraic number field $k$ and $X(k)$ the set of $k$-rational points of $X$. We fix some "adelic space" $Y$ which contains $X(k)$, for example, one can take $X(A)$ as $Y$ if $X$ is an affine variety. We want to study the distribution of $X(k)$ in $Y$. To do this, we take a "height function" $H$ on $Y$ and define the "ball" $B_T$ of radius $T > 0$ in $Y$ as the set of points $y$ with $H(y) \leqq T$. By the Hardy- -Littlewood method, one can expect that the number of $k$-rational points in $B_T$ is asymptotically equal to the "volume" of $B_T$ as $T$ tends to infinity. We have two examples of a family of homogeneous spaces which satisfy this property. They are (A) Affine symmetric spaces and (P) Flag varieties. In this talk, we will show the Hardy--Littlewood property of the case (P). In the case (P), moreover, we can describe the distribution of $X(k)$ around "the origin", i.e., we can evaluate the radius $T$ such that $B_T$ has no $k$-rational point. This result leads us to the notion of "Hermite's constant of $X$".

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