Reference Request: Anisotropic Algebraic Groups Have No Unipotent Elements

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I have found the following fact stated in a number of places:



If $k$ is any field, a connected reductive group $G$ is anisotropic if and only if its only unipotent element is $e$ and $mathrmHom_k(G, mathrmG_m)$ is trivial.



For instance, this appears in section 3.4 of Springer's Corvallis article. However, I have been unable to track down a reference for a proof of this result.






reference-request algebraic-groups







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    up vote
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    I have found the following fact stated in a number of places:



    If $k$ is any field, a connected reductive group $G$ is anisotropic if and only if its only unipotent element is $e$ and $mathrmHom_k(G, mathrmG_m)$ is trivial.



    For instance, this appears in section 3.4 of Springer's Corvallis article. However, I have been unable to track down a reference for a proof of this result.






    reference-request algebraic-groups







    share|cite|improve this question























      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      I have found the following fact stated in a number of places:



      If $k$ is any field, a connected reductive group $G$ is anisotropic if and only if its only unipotent element is $e$ and $mathrmHom_k(G, mathrmG_m)$ is trivial.



      For instance, this appears in section 3.4 of Springer's Corvallis article. However, I have been unable to track down a reference for a proof of this result.






      reference-request algebraic-groups







      share|cite|improve this question













      I have found the following fact stated in a number of places:



      If $k$ is any field, a connected reductive group $G$ is anisotropic if and only if its only unipotent element is $e$ and $mathrmHom_k(G, mathrmG_m)$ is trivial.



      For instance, this appears in section 3.4 of Springer's Corvallis article. However, I have been unable to track down a reference for a proof of this result.






      reference-request algebraic-groups




      reference-request algebraic-groups






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      asked 2 hours ago









      Alexander

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      33416




















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          Corollary 8.5 of Borel -Tits proves this in characteristic zero:



          http://www.numdam.org/item?id=PMIHES_1965__27__55_0



          See also section 4 of the same article (where other fields are considered, but it is not said in terms of unipotent elements, because that is false). Indeed, the group $PGL_1(D)$ over a field $k$ of positive characteristic $p$, $D$ a central division algebra over $k$ of degree $p$ , can have "bad" unipotent elements coming from purely inseparable extensions of $k$ lying in $D$. What is true is that $G$ is anisotropic iff it does not have a proper parabolic subgroup defined over $k$.



          In any case, this is the standard reference for reductive groups over arbitrary fields






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            1 Answer
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            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

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            up vote
            3
            down vote



            accepted










            Corollary 8.5 of Borel -Tits proves this in characteristic zero:



            http://www.numdam.org/item?id=PMIHES_1965__27__55_0



            See also section 4 of the same article (where other fields are considered, but it is not said in terms of unipotent elements, because that is false). Indeed, the group $PGL_1(D)$ over a field $k$ of positive characteristic $p$, $D$ a central division algebra over $k$ of degree $p$ , can have "bad" unipotent elements coming from purely inseparable extensions of $k$ lying in $D$. What is true is that $G$ is anisotropic iff it does not have a proper parabolic subgroup defined over $k$.



            In any case, this is the standard reference for reductive groups over arbitrary fields






            share|cite|improve this answer


























              up vote
              3
              down vote



              accepted










              Corollary 8.5 of Borel -Tits proves this in characteristic zero:



              http://www.numdam.org/item?id=PMIHES_1965__27__55_0



              See also section 4 of the same article (where other fields are considered, but it is not said in terms of unipotent elements, because that is false). Indeed, the group $PGL_1(D)$ over a field $k$ of positive characteristic $p$, $D$ a central division algebra over $k$ of degree $p$ , can have "bad" unipotent elements coming from purely inseparable extensions of $k$ lying in $D$. What is true is that $G$ is anisotropic iff it does not have a proper parabolic subgroup defined over $k$.



              In any case, this is the standard reference for reductive groups over arbitrary fields






              share|cite|improve this answer
























                up vote
                3
                down vote



                accepted







                up vote
                3
                down vote



                accepted






                Corollary 8.5 of Borel -Tits proves this in characteristic zero:



                http://www.numdam.org/item?id=PMIHES_1965__27__55_0



                See also section 4 of the same article (where other fields are considered, but it is not said in terms of unipotent elements, because that is false). Indeed, the group $PGL_1(D)$ over a field $k$ of positive characteristic $p$, $D$ a central division algebra over $k$ of degree $p$ , can have "bad" unipotent elements coming from purely inseparable extensions of $k$ lying in $D$. What is true is that $G$ is anisotropic iff it does not have a proper parabolic subgroup defined over $k$.



                In any case, this is the standard reference for reductive groups over arbitrary fields






                share|cite|improve this answer














                Corollary 8.5 of Borel -Tits proves this in characteristic zero:



                http://www.numdam.org/item?id=PMIHES_1965__27__55_0



                See also section 4 of the same article (where other fields are considered, but it is not said in terms of unipotent elements, because that is false). Indeed, the group $PGL_1(D)$ over a field $k$ of positive characteristic $p$, $D$ a central division algebra over $k$ of degree $p$ , can have "bad" unipotent elements coming from purely inseparable extensions of $k$ lying in $D$. What is true is that $G$ is anisotropic iff it does not have a proper parabolic subgroup defined over $k$.



                In any case, this is the standard reference for reductive groups over arbitrary fields







                share|cite|improve this answer














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                edited 1 hour ago

























                answered 1 hour ago









                Venkataramana

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