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://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_
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%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 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 g12(qq12)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=p2p. 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