Rational Points on Curves over Finite Fields Part I : « q large » - Part II :  « g large »
Cours de Jean-Pierre Serre, no. 6 (1985) , 242 p.
@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://archive.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://archive.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://archive.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_/

Table of contents

Part I - « q large » p. ii
Contents – Part Ip. 2
Introductionp. 4
Weil boundp. 4
Connection to codesp. 6
General Resultsp. 9
Refined Weil bound p. 9
Refinements using traces of algebraic integersp. 14
Smyth’s proof of Siegel’s Theoremp. 21
Indecomposability of jacobians and applicationsp. 23
Beauville’s Theoremp. 27
The case g=1 (review)p. 32
The case g=2p. 39
Results of Tate and Honda and applicationsp. 39
« Glueing » elliptic curvesp. 45
Statement of Theoremp. 49
Remarks on « special » qp. 50
The elementary glueingp. 53
Proof for q a squarep. 60
Intermezzo of proof for q a squarep. 64
Conclusion of proof for q a squarep. 70
Proof for q not a square, not specialp. 77
Proof for q not a square, specialp. 83
« Glueing » and Hermitian modulesp. 98
Proof using hermitian modulesp. 106
The Skolem method for diophantine equationsp. 121
The case g=3p. 128
Voloch’s boundp. 128
Constructing curves : some examplesp. 132
Conjecturesp. bis 139
Part II - « g large » p. 143
Contents – Part IIp. 144
General Resultsp. 145
The bound g1 2(qq 1 2 )p. 145
When is the Weil bound attained ?p. 147
Explicit formula and applicationsp. 149
Asymptotic results as gp. 160
Ihara’s theorem on towersp. 163
Modular Curves and the case q=p 2 p. 165
Class Field Towersp. ter 170
The Theorem of Golod and Šafarevičp. ter 170
Infinite class field towersp. 176
Construction in characteristic p2p. 183
Construction for q=2p. 191
Something from Class Field Theoryp. 196
Optimal bounds from the explicit formulap. 202
The optimization problem and its dualp. 202
Oesterlé’s Theoremp. 208
The case q=2 p. 221
Boundsp. 221
Construction of curvesp. 227