Adeles and Tamagawa numbers - Tome (1981) no. 1

Adeles and Tamagawa numbers

Serre, Jean-Pierre

Cours de Jean-Pierre Serre, no. 1 (1981) , 204 p.

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@book{CJPS_1981__1_,
     author = {Serre, Jean-Pierre},
     title = {Adeles and {Tamagawa} numbers},
     series = {Cours de Jean-Pierre Serre},
     publisher = {Harvard},
     number = {1},
     year = {1981},
     language = {en},
     url = {http://archive.numdam.org/item/CJPS_1981__1_/}
}

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AU  - Serre, Jean-Pierre
TI  - Adeles and Tamagawa numbers
T3  - Cours de Jean-Pierre Serre
PY  - 1981
IS  - 1
PB  - Harvard
UR  - http://archive.numdam.org/item/CJPS_1981__1_/
LA  - en
ID  - CJPS_1981__1_
ER  -

%0 Book
%A Serre, Jean-Pierre
%T Adeles and Tamagawa numbers
%S Cours de Jean-Pierre Serre
%D 1981
%N 1
%I Harvard
%U http://archive.numdam.org/item/CJPS_1981__1_/
%G en
%F CJPS_1981__1_

Serre, Jean-Pierre. Adeles and Tamagawa numbers. Cours de Jean-Pierre Serre, no. 1 (1981), 204 p. http://numdam.org/item/CJPS_1981__1_/

Sommaire

History	p. 1
Siegel formula	p. 3
Tamagawa	p. 8
I - Integration on $p$ -adic manifolds page	p. 9
Notation	p. 9
Measure attached to a form $ω$	p. 10
Rational summation	p. 15
Igusa	p. 16
Vector bundles	p. 17
Smooth schemes	p. 18
Number of points of classical and exceptional groups	p. 20
Decomposition of a measure	p. 24
Connection with densities	p. 28
Digression : Liftable solutions	p. 31
Smooth case	p. 32
Real case	p. 37
Use of resolution of singularities	p. 40
Oesterlé : hypersurfaces	p. 41
Łojasiewicz inequality	p. 48
Oesterlé : general case	p. 49
II - Adeles	p. 55
History	p. 55
Definition	p. 55
$𝔸_{K} / K$ compact	p. 58
Haar measure on $𝔸_{K}$	p. 61
Characters and duality	p. 64
Haar measure for dual groups	p. 66
Compatible measures with respect to duality	p. 67
Adelic integration and heuristic formulas (Goldbach,...)	p. 69
Adelic points of algebraic varieties	p. 75
Properties of the functor $V \mapsto V (𝔸_{K})$	p. 78
Restriction of scalars	p. 84
Algebraic groups and adelic points	p. 88
Abelian varieties (S. Bloch)	p. 88
Weak approximation	p. 90
Strong approximation	p. 91
Adeles, classes and genera	p. 99
Tensors page	p. 102
Vector bundles	p. 104
Adelic measures	p. 106
Convergent case	p. 108
Algebraic groups	p. 111
Tori	p. 114
Convergence	p. 117
Tamagawa number (semi-simple case)	p. 122
Theorem of Ono – Weil conjecture	p. 123
Tamagawa number (reductive case)	p. 126
$τ (𝔾_{m}) = 1$	p. 131
Tori (Ono)	p. 133
$τ (P G L_{n}) = n τ (S L_{n})$	p. 135
Tamagawa $\leftrightarrow$ Siegel (mass formula)	p. 136
Tamagawa $\leftrightarrow$ Siegel (the two-groups game)	p. 145
Correction to Ono's theory	p. 146
Positive definite quadratic forms over $ℚ$	p. 147
Proof that $τ (S O_{n}) = 2$ for $n = 2, 3, 4$	p. 152
Proof of Siegel's formula	p. 156
Proof ( $m \geq 5$ )	p. 163
Proof ( $m = 4$ )	p. 165
Proof ( $m = 2$ )	p. 166
Proof ( $m = 3$ )	p. 169
Remarks on Siegel's proof	p. 172
Application to modular forms	p. 176
III - $S L_{n}$	p. 180
The Minkowski-Hlawka theorem	p. 180
Proof	p. 184