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Geometry of numbers in adele spaces

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Rozprawy Matematyczne tom/nr w serii: 88 wydano: 1971
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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
Słowa kluczowe
Tematy
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne tom/nr w serii: 88
Liczba stron
49
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom LXXXVIII
Daty
wydano
1971
Twórcy
autor
Bibliografia
  • [1] R. P. Bambah, A. Woods and H. Zassenhaus, Three proofs of Minkowski's second inequality in the geometry of numbers, J. Austral. Math. Soc. 5 (1965), pp. 453-462.
  • [2] D. G. Cantor, On the elementary theory of diophantine approximation over the ring of adeles I, Illinois J. Math. 9 (1965), pp. 677-700.
  • [3] J. W. S. Cassets, An introduction to the geometry of numbers, Berlin 1959.
  • [4] C. Chabauty, Limite d'ensembles et géométrie des nombres, Bull. Soc. Math. France 77 (1950), pp. 143-151.
  • [5] S. Lang, Algebraic numbers, Addison-Wesley, Reading, Maes., 1964.
  • [6] I. S. Luthar, A note on a result of Mahler's, J. Austral. Math. Soc. 6 (1966), pp. 399-401.
  • [7] E. Lutz, Sur les approximations diophantiehnes linéaires P-adiques, Actualités Sci. Ind. 1224, Paris 1955.
  • [8] A. M. Macbeath, Abstract theory of packings and coverings I, Proc. Glasgow Math. Assoc. 4 (1959-1960), pp. 92-95.
  • [9] A. M. Macbeath and S. Swierczkowski, Limits of lattices in a compactly generated group, Canadian J. Math. 12 (1960), pp. 427-437.
  • [10] K. Mahler, An analogue to Minkowski's geometry of numbers in a field of series, Ann. of Math. 42 (1941), pp. 488-522.
  • [11] K. Mahler, Inequalities for ideal bases in algebraic number fields, J. Austral. Math. Soc. 4 (1964), pp. 425-448.
  • [12] R. B. McFeat, Geometry of numbers in adele spaces, University of Adelaide Ph. D. thesis, 1969.
  • [13] K. Rogers and H. P. F. Swinnerton-Dyer, Geometry of numbers over algebraic number fields, Trans. Amer. Math. Soc. 88 (1958), pp. 227-242.
  • [14] S. Swierczkowski, Abstract theory of packings and coverings II, Proc. Glasgow Math. Assoc. 4 (1959-1960), pp. 96-100.
  • [15] J. T. Tate, Fourier analysis in number fields and Hecke's zeta-functions, Algebraic number theory (Proc. Instructional Conference, Brighton, 1965), pp. 305-347, London 1967.
  • [16] E. Weiss, Algebraic number theory, New York 1963.
  • [17] J. W. S. Cassels, Global fields, Algebraic number theory (Proc. Instructional Conference, Brighton 1965), pp. 42-84, London 1967.
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