Return to Answer

In addition to Jason's answer, I mention the following result, which I found out to be not known to experts (except Jason).

Theorem. Let $X$ be a homogeneous space of a connected linear algebraic group $G$ over a field $k$, with connected geometric stabilizer. Assume that $X$ has a zero cycle of degree 1. If $k$ is a either a $p$-adic field or a number field, then $X$ has a $k$-point.

I give a proof based on Jason's observation (actually the case of a $p$-adic field is contained in his answer) and use the paper by Borovoi, Colliot-Thélène and Skorobogatov [BCS] that Jason cites.

Proof. If $X$ has a zero cycle of degree 1, then the elementary obstruction for $X$ is 0. If $k$ is a $p$-adic field, then by [BCS], Thm. 3.3, $X$ has a $k$-point. If $k$ is a number field, then for any real place $v$ of $k$, $X$ has a zero cycle of degree 1 over $k_v$, hence $X$ has a $k_v$-point (because $k_v$ is isomorphic to $\mathbf{R}$), and by [BCS], Thm. 3.10, $X$ has a $k$-point.

Another proof of this theorem was recently obtained by Cyril Demarche and Liang Yongqi.

Note that Jodi Black http://arxiv.org/abs/1010.1582 recently proved that if a principal homogeneous space $X$ of a connected linear group $G$ over a field $k$ of virtual cohomological dimension $\le 2$ has a zero cycle of degree 1, and $G$ satisfies the Hasse principle, then $X$ has a $k$-point.

In addition to Jason's answer, I mention the following result, which I found out to be not known to experts (except Jason).

Theorem. Let $X$ be a homogeneous space of a connected linear algebraic group $G$ over a field $k$, with connected geometric stabilizers. Assume that $X$ has a zero cycle of degree 1. If $k$ is a either a $p$-adic field or a number field, then $X$ has a $k$-point.

I give a proof based on Jason's observation (actually the case of a $p$-adic field is contained in his answer) and use the paper by Borovoi, Colliot-Thélène and Skorobogatov [BCS] that Jason cites.

Proof. If $X$ has a zero cycle of degree 1, then the elementary obstruction for $X$ is 0. If $k$ is a $p$-adic field, then by [BCS], Thm. 3.3, $X$ has a $k$-point. If $k$ is a number field, then for any real place $v$ of $k$, $X$ has a zero cycle of degree 1 over $k_v$, hence $X$ has a $k_v$-point (because $k_v$ is isomorphic to $\mathbf{R}$), and by [BCS], Thm. 3.10, $X$ has a $k$-point.

Another proof of this theorem was recently obtained by Cyril Demarche and Liang Yongqi.

Note that both assumptions of the theorem, namely that geometric stabilizers are connected and that the base field $k$ is either a $p$-adic field or a number field, are important.

Mathieu Florence in the paper Zéro-cycles de degré un sur les espaces homogènes, Int. Math. Res. Not. 2004, no. 54, 2897–2914, http://alg-geo.epfl.ch/~florence/esphomog.pdf, constructed homogeneous spaces $X$ over $p$-adic and number fields with non-connected (finite) geometric stabilizers, such that $X$ has a zero cycle of degree 1, but neither $X$ nor any smooth compactification of $X$ has rational points.

Parimala in the paper Homogeneous varieties — zero cycles of degree one versus rational points, Asian J. Math. 9 (2005), 251–256, see the link in Artie's answer, constructed a projective homogeneous space $X$ (hence with connected geometric stabilizers) over the Laurent series field over a $p$-adic field, such that again $X$ has a zero cycle of degree 1, but no rational points.

Note that Jodi Black http://arxiv.org/abs/1010.1582 recently proved that if a principal homogeneous space $X$ of a connected linear group $G$ over a field $k$ of virtual cohomological dimension $\le 2$ has a zero cycle of degree 1, and $G$ satisfies the Hasse principle, then $X$ has a $k$-point.

In addition to Jason's answer, I mention the following result, which I found out to be not known to experts (except Jason).

Theorem. Let $X$ be a homogeneous space of a connected linear algebraic group $G$ over a field $k$, with connected geometric stabilizer. Assume that $X$ has a zero cycle of degree 1. If $k$ is a either a $p$-adic field or a number field, then $X$ has a $k$-point.

I give a proof based on Jason's observation (actually the case of a $p$-adic field is contained in his answer) and use the paper by Borovoi, Colliot-Thélène and Skorobogatov [BCS] that Jason cites.

Proof. If $X$ has a zero cycle of degree 1, then the elementary obstruction for $X$ is 0. If $k$ is a $p$-adic field, then by [BCS], Thm. 3.3, $X$ has a $k$-point. If $k$ is a number field, then for any real place $v$ of $k$, $X$ has a zero cycle of degree 1 over $k_v$, hence $X$ has a $k_v$-point (because $k_v$ is isomorphic to $\mathbf{R}$), and by [BCS], Thm. 3.10, $X$ has a $k$-point.

Another proof of this theorem was recently obtained by Cyril Demarche and Liang Yongqi.

Note that Jodi Black http://arxiv.org/abs/1010.1582 recently proved that if a principal homogeneous space $X$ of a connected linear group $G$ over a field $k$ of virtual cohomological dimension $\le 2$ has a zero cycle of degree 1, and $G$ satisfies the Hasse principle, then $X$ has a $k$-point.

In addition to Jason's answer, I mention the following result, which I found out to be not known to experts (except Jason).

Theorem. Let $X$ be a homogeneous space of a connected linear algebraic group $G$ over a field $k$, with connected geometric stabilizer. Assume that $X$ has a zero cycle of degree 1. If $k$ is a either a $p$-adic field or a number field, then $X$ has a $k$-point.

I give a proof based on Jason's observation (actually the case of a $p$-adic field is contained in his answer) and use the paper by Borovoi, Colliot-Thélène and Skorobogatov [BCS] that Jason cites.

Proof. If $X$ has a zero cycle of degree 1, then the elementary obstruction for $X$ is 0. If $k$ is a $p$-adic field, then by [BCS], Thm. 3.3, $X$ has a $k$-point. If $k$ is a number field, then for any real place $v$ of $k$, $X$ has a zero cycle of degree 1 over $k_v$, hence $X$ has a $k_v$-point (because $k_v$ is isomorphic to $\mathbf{R}$), and by [BCS], Thm. 3.10, $X$ has a $k$-point.

Another proof of this theorem was recently obtained by Cyril Demarche and Liang Yongqi.

Note that Jodi Black http://arxiv.org/abs/1010.1582 recently proved that if a principal homogeneous space $X$ of a connected linear group $G$ over a field $k$ of virtual cohomological dimension $\le 2$ has a zero cycle of degree 1, and $G$ satisfies the Hasse principle, then $X$ has a $k$-point.

In addition to Jason's answer, I mention the following result, which I found out to be not known to experts (except Jason).

Theorem. Let $X$ be a homogeneous space of a connected linear algebraic group $G$ over a field $k$, with connected geometric stabilizer. Assume that $X$ has a zero cycle of degree 1. If $k$ is a either a $p$-adic field or a number field, then $X$ has a $k$-point.

I give a proof based on Jason's observation (actually the case of a $p$-adic field is contained in his answer) and use the paper by Borovoi, Colliot-Thélène and Skorobogatov [BCS] that Jason cites.

Proof. If $X$ has a zero cycle of degree 1, then the elementary obstruction for $X$ is 0. If $k$ is a $p$-adic field, then by [BCS], Thm. 3.3, $X$ has a $k$-point. If $k$ is a number field, then for any real place $v$ of $k$, $X$ has a zero cycle of degree 1 over $k_v$, hence $X$ has a $k_v$-point (because $k_v$ is isomorphic to $\mathbf{R}$), and by [BCS], Thm. 3.10, $X$ has a $k$-point.

Another proof of this theorem was recently obtained by Cyril Demarche and Liang Yongqi.

Note that Jodi Black http://arxiv.org/abs/1010.1582 recently proved that if a principal homogeneous space $X$ of a connected linear group $G$ over a field $k$ of virtual cohomological dimension $\le 2$ has a zero cycle of degree 1, and $G$ satisfies the Hasse principle, then $X$ has a $k$-point.

Let somebody who masters Markdown edit my answer!

In addition to Jason's answer, I mention the following result, which I found out to be not known to experts (except Jason).

Theorem. Let $X$ be a homogeneous space of a connected linear algebraic group $G$ over a field $k$, with connected geometric stabilizer. Assume that $X$ has a zero cycle of degree 1. If $k$ is a either a $p$-adic field or a number field, then $X$ has a $k$-point.

I give a proof based on Jason's observation (actually the case of a $p$-adic field is contained in his answer) and use the paper by Borovoi, Colliot-Thélène and Skorobogatov [BCS] that Jason cites.

Proof. If $X$ has a zero cycle of degree 1, then the elementary obstruction for $X$ is 0. If $k$ is a $p$-adic field, then by [BCS], Thm. 3.3, $X$ has a $k$-point. If $k$ is a number field, then for any real place $v$ of $k$, $X$ has a zero cycle of degree 1 over $k_v$, hence $X$ has a $k_v$-point (because $k_v$ is isomorphic to $\mathbf{R}$), and by [BCS], Thm. 3.10, $X$ has a $k$-point.

Another proof of this theorem was recently obtained by Cyril Demarche and Liang Yongqi.

Note that Jodi Black http://arxiv.org/abs/1010.1582 recently proved that if a principal homogeneous space $X$ of a connected linear group $G$ over a field $k$ of virtual cohomological dimension $\le 2$ has a zero cycle of degree 1, and $G$ satisfies the Hasse principle, then $X$ has a $k$-point.