Rational Points on Curves over Finite Fields Part I : « $q$ large » - Part II : « $g$ large »

Rational Points on Curves over Finite Fields Part I : «

q

large » - Part II : «

g

large »

Serre, Jean-Pierre

Notes prises par : Gouvêa, Fernando Q.

Cours de Jean-Pierre Serre, no. 6 (1985) , 242 p.

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@book{CJPS_1985__6_,
     author = {Serre, Jean-Pierre},
     title = {Rational {Points} on {Curves} over {Finite} {Fields} {Part} {I~:} {\guillemotleft}~$q$ large~{\guillemotright} - {Part} {II~:~} {\guillemotleft}~$g$ large~{\guillemotright}},
     series = {Cours de Jean-Pierre Serre},
     number = {6},
     year = {1985},
     language = {en},
     url = {http://www.numdam.org/item/CJPS_1985__6_/}
}

TY  - BOOK
AU  - Serre, Jean-Pierre
TI  - Rational Points on Curves over Finite Fields Part I : « $q$ large » - Part II :  « $g$ large »
T3  - Cours de Jean-Pierre Serre
PY  - 1985
IS  - 6
UR  - http://www.numdam.org/item/CJPS_1985__6_/
LA  - en
ID  - CJPS_1985__6_
ER  -

%0 Book
%A Serre, Jean-Pierre
%T Rational Points on Curves over Finite Fields Part I : « $q$ large » - Part II :  « $g$ large »
%S Cours de Jean-Pierre Serre
%D 1985
%N 6
%U http://www.numdam.org/item/CJPS_1985__6_/
%G en
%F CJPS_1985__6_

Serre, Jean-Pierre. Rational Points on Curves over Finite Fields Part I : « $q$ large » - Part II :  « $g$ large ». Cours de Jean-Pierre Serre, no. 6 (1985), Gouvêa, Fernando Q. (red.), 242 p. http://numdam.org/item/CJPS_1985__6_/

Sommaire

Part I - « $q$ large »	p. ii
Contents – Part I	p. 2
Introduction	p. 4
Weil bound	p. 4
Connection to codes	p. 6
General Results	p. 9
Refined Weil bound	p. 9
Refinements using traces of algebraic integers	p. 14
Smyth’s proof of Siegel’s Theorem	p. 21
Indecomposability of jacobians and applications	p. 23
Beauville’s Theorem	p. 27
The case $g = 1$ (review)	p. 32
The case $g = 2$	p. 39
Results of Tate and Honda and applications	p. 39
« Glueing » elliptic curves	p. 45
Statement of Theorem	p. 49
Remarks on « special » $q$	p. 50
The elementary glueing	p. 53
Proof for $q$ a square	p. 60
Intermezzo of proof for $q$ a square	p. 64
Conclusion of proof for $q$ a square	p. 70
Proof for $q$ not a square, not special	p. 77
Proof for $q$ not a square, special	p. 83
« Glueing » and Hermitian modules	p. 98
Proof using hermitian modules	p. 106
The Skolem method for diophantine equations	p. 121
The case $g = 3$	p. 128
Voloch’s bound	p. 128
Constructing curves : some examples	p. 132
Conjectures	p. bis 139
Part II - « $g$ large »	p. 143
Contents – Part II	p. 144
General Results	p. 145
The bound $g \leq \frac{1}{2} (q - q^{\frac{1}{2}})$	p. 145
When is the Weil bound attained ?	p. 147
Explicit formula and applications	p. 149
Asymptotic results as $g \to \infty$	p. 160
Ihara’s theorem on towers	p. 163
Modular Curves and the case $q = p^{2}$	p. 165
Class Field Towers	p. ter 170
The Theorem of Golod and Šafarevič	p. ter 170
Infinite class field towers	p. 176
Construction in characteristic $p \neq 2$	p. 183
Construction for $q = 2$	p. 191
Something from Class Field Theory	p. 196
Optimal bounds from the explicit formula	p. 202
The optimization problem and its dual	p. 202
Oesterlé’s Theorem	p. 208
The case $q = 2$	p. 221
Bounds	p. 221
Construction of curves	p. 227