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Geometry of numbers in adele spaces

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McFeat R. B.

Instytut Matematyczny Polskiej Akademii Nauk

CONTENTS Part I 1. Introduction...................................................................................................................................................... 5 2. Preliminaries.................................................................................................................................................. 6 2.1. Notation........................................................................................................................................................ 6 2.2. Local preliminaries.................................................................................................................................... 6 3. The space $\mathscr{A}$............................................................................................................................. 9 3.1. Generalities................................................................................................................................................. 9 3.2. Linear transformations of $\mathscr{A}$............................................................................................... 10 3.3. Global measure.......................................................................................................................................... 11 4. Lattices and convex bodies.......................................................................................................................... 11 4.1. Lattices......................................................................................................................................................... 11 4.2. Convex bodies............................................................................................................................................. 13 5. An analogue of Minkowski's convex body theorem................................................................................. 15 5.1. Convex body theorem................................................................................................................................ 15 5.2. Applications of theorem 2......................................................................................................................... 16 6. Successive minima....................................................................................................................................... 18 6.1. Preliminaries............................................................................................................................................... 18 6.2. The product of successive minima; an upper bound.......................................................................... 19 6.3. The product of successive minima; a lower bound............................................................................ 22 6.4. Applications to algebraic number theory................................................................................................ 24 7. T-adeles........................................................................................................................................................... 32 7.1. The general theory for T-adeles............................................................................................................... 32 7.2. Two special cases..................................................................................................................................... 35 Part II 1. Introduction ..................................................................................................................................................... 37 2. Topology in $\mathscr{G}$................................................................................................................... 37 2.1. Two topologies on $\mathscr{G}$........................................................................................................... 37 2.2. Comparison of the two topologies.......................................................................................................... 39 3. Compactness for lattices............................................................................................................................. 41 3.1. Two topologies on the lattice space....................................................................................................... 41 3.2. An important lemma................................................................................................................................... 43 3.3. An analogue of Mahler’s compactness theorem................................................................................. 44 4. The Chabauty topology................................................................................................................................. 45 5. T-adeles ......................................................................................................................................................... 47 References.......................................................................................................................................................... 49

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1 McFeat R. B.

1 1971

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Geometry of numbers in adele spaces