# Finiteness theorems for algebraic groups over function fields

# Finiteness theorems for algebraic groups over function fields

If $X$ is a smooth variety over a global field $k$, $G$ is an algebraic group over $k$ equipped with an action on $X$, and $x$ is a point in $X(k)$ then it is natural to ask how the property of $x'$ in $X(k)$ being in the $G(k)$-orbit of $x$ compares with being in the $G(k_v)$-orbit of $x$ for all places $v$ of $k$. In general there is a non-trivial "local-to-global" obstruction space, but one can ask if it is finite. Even when $G$ is semisimple, this finiteness problem leads to the consideration of the isotropy group $G_x$ that is generally not connected or reductive (or even smooth when $char(k)>0$). In the number field case the finiteness of these obstruction spaces was proved by Borel and Serre long ago, but their method used characteristic $0$ in an essential way. Recently in joint work with Gabber and G. Prasad we have developed a theory of "pseudo-reductive groups" which is a very useful tool to prove results for general affine algebraic groups in the function field case that were previously known only in the reductive case. In particular, this work makes it possible to prove the analogue of the Borel-Serre finiteness result over function fields (away from char. 2 for now). The first part of the talk will explain a bit about the theory of pseudo-reductive groups, and the rest of the talk will show how it is used to establish the finiteness of the local-to-global obstruction spaces in the function field case (in $char>2$). If time permits we will also discuss an application to the problem of whether the $k$-isomorphism class of a projective $k$-variety is determined (up to "finite ambiguity") by its $k_v$-isomorphism class for all places $v$ of $k$ (a problem solved by Mazur over number fields, once again making essential use of characteristic $0$).