## Algebraic Groups and Their Birational InvariantsSince the late 1960s, methods of birational geometry have been used successfully in the theory of linear algebraic groups, especially in arithmetic problems. This book studies birational properties of linear algebraic groups focusing on arithmetic applications. |

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### Contents

Birational Geometry of Algebraic Tori | 41 |

Invariants of Finite Transformation Groups | 75 |

Arithmetic of Linear Algebraic Groups | 103 |

Tamagawa Numbers | 133 |

i?equivalence in Algebraic Groups | 177 |

197 | |

Bibliographical Remarks | 207 |

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### Common terms and phrases

5-scheme abelian group adjoint affine algebraic tori algebraic torus algebraic variety algebraically closed arbitrary automorphism basis bijection birational invariant birationally equivalent Brnr called character characteristic zero Chevalley compute connected linear algebraic Consider corresponding decomposition defined denote direct product direct sum divisors dual elements embedding epimorphism exact sequence Example fc-algebra fc-form fc-group fc-rational fc-scheme fc-torus field of invariants field of rational finite group finite type flasque resolution formula functor Furthermore Gal(L/fc Galois extension Galois group GL(n group G group scheme hence homomorphism ideal II-module implies integer involution irreducible isomorphism kernel lattice Let G linear algebraic group matrix maximal torus module morphism number field permutation PicX polynomial prime Proof quadratic quasisplit quotient rational functions reduction representation ring semisimple group shows simply connected group space Spec splitting field stably rational subset Theorem topology torsor trivial unipotent Weyl group