Geometry of numbers in adele spaces

R. B. McFeat

GDML_Books, (1971), p.

Résumé

CONTENTSPart I1. Introduction...................................................................................................................................................... 52. Preliminaries.................................................................................................................................................. 62.1. Notation........................................................................................................................................................ 62.2. Local preliminaries.................................................................................................................................... 63. The space $\mathcal {A}$............................................................................................................................. 93.1. Generalities................................................................................................................................................. 93.2. Linear transformations of $\mathcal {A}$............................................................................................... 103.3. Global measure.......................................................................................................................................... 114. Lattices and convex bodies.......................................................................................................................... 114.1. Lattices......................................................................................................................................................... 114.2. Convex bodies............................................................................................................................................. 135. An analogue of Minkowski’s convex body theorem................................................................................. 155.1. Convex body theorem................................................................................................................................ 155.2. Applications of theorem 2......................................................................................................................... 166. Successive minima....................................................................................................................................... 186.1. Preliminaries............................................................................................................................................... 186.2. The product of successive minima; an upper bound.......................................................................... 196.3. The product of successive minima; a lower bound............................................................................ 226.4. Applications to algebraic number theory................................................................................................ 247. T-adeles........................................................................................................................................................... 327.1. The general theory for T-adeles............................................................................................................... 327.2. Two special cases..................................................................................................................................... 35Part II1. Introduction ..................................................................................................................................................... 372. Topology in $\mathcal {G}$................................................................................................................... 372.1. Two topologies on $\mathcal {G}$........................................................................................................... 372.2. Comparison of the two topologies.......................................................................................................... 393. Compactness for lattices............................................................................................................................. 413.1. Two topologies on the lattice space....................................................................................................... 413.2. An important lemma................................................................................................................................... 433.3. An analogue of Mahler’s compactness theorem................................................................................. 444. The Chabauty topology................................................................................................................................. 455. T-adeles ......................................................................................................................................................... 47References.......................................................................................................................................................... 49

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