椭圆函数

所属分类:数学  
出版时间:2003-11   出版时间:北京世界图书出版公司   作者:S.Lang   页数:326  
Tag标签:GTM,Springer,数学  

内容概要

Elliptic functions parametrize elliptic curves, and the intermingling of the analytic and algebraic-arithmetic theory has been at the center of mathematics since the early part of the nineteenth century. Some new techniques and outlooks have recently appeared on these old subjects, continuing in the tradition of Kronecker, Weber, Fricke, Hasse,Deuring. Shimura's book Introduction to the arithmetic theory of automorphic functions is a splendid modern reference, which I found very helpful myself to learn some aspects of elliptic curves. It emphasizes the direction of the Hasse-Weil zeta function, Hecke operators, and the generalizations due to him to the higher dimensional case (abelian varieties, curves of higher genus coming from an arithmetic group operating on the upper half plane, bounded symmetric domains with a discrete arithmetic group whose quotient is algebraic). I refer the interested reader to his book and the bibliography therein.

书籍目录

PART
ONEccGENERAL
THEORY
 Chapter1
Ellipti
Functions
  1
ThecLiouville
Theorems
  2
The
Weierstrass
Function
  3
The
AdditioncTheorem
  4
Isomorphism
Classescof
Elliptic
Curves
  5
Endomorphisms
and
Automorphisms
  Chapter2
Homomorphisms
  1
Points
of
Finite
Order
  2
Isogenies
  3
The
Involution
 Chapter
3
hecModular
Function
  1
The
Modular
Group
  2
Automorphic
Functions
of
Degree
2k
  3
The
Modular
Functionj
 Chapter
4
Fourier
Expansions
  1
Expansion
for
Gk,cg2,cg3,c△candcj
  2
Expansion
for
the
Weierstrass
Function
  3
Bernoulli
Numbers
 Chapter
5
The
Modular
Equation
  1
Integral
Matrices
with
Positive
Determinant
  2
The
Modular
Equation
  3
Relations
with
Isogenies
 Chapter
6
Higher
Levels
  1
Congruence
Subgroups
  2
The
Field
of
Modular
Functions
OvercC
  3
The
Field
of
Modular
Functions
OvercQ
  4
Subfields
of
the
Modular
Function
Field
 Chapter
7
Automorphisms
of
the
Modular
Function
Field
  1
Rational
Adeles
of
GL
  2
Operation
of
the
Rational
Adelescon
the
Modular
Function
Field
  3
The
Shimura
Exact
Sequence
  PARTcTWOccCOMPLEXcMULTIPLICATION
ELLIPTICcCURVEScWITHcSINGULARcINVARIANTS
 Chapter
8
Results
from
Algebraic
Number
Theory
  1
Latticescin
Quadratic
Fields
  2
Completions
  3
The
Decomposition
Group
and
Frobenius
Automorphism
  4
Summary
of
Class
Field
Theory
Chapter
9
Reduction
of
Elliptic
Curves
 
1
Non-degenerate
Reduction,
General
Case
 
2
Redu
tion
of
Homomorphisms
  3
Coverings
of
LevelcN
 
4
Reduction
of
Differential
Forms
 
Chapter
10
Complex
Multiplication
 
1
Generation
of
Class
Fields,
Deuring's
Approach
  2
Idelic
Formulation
for
Arbitrary
Lattices
 
3
Generation
of
Class
Fields
by
Singular
Values
of
Modular
Functions
  4
The
Frobenius
Endomorphism
 
Appendix
A
Relation
of
Kronecker
Chapter
11
Shimura's
Reciprocity
Law
 
I
Relation
Between
Generic
and
Special
Extensions
 
2
Application
to
Quotientscof
Modular
Forms
Chapter
12
The
Fun
tion
△(at)/△(t)
  1
Behavior
Under
the
Artin
Automorphism
  2
Prime
Factorization
of
its
Values
  3
Analyti
Proof
for
the
Congruence
Relationcofj
 Chapterc13
The
l-adic
and
p-adic
Representations
of
Deuring
1
Thecl-adic
Spaces
2
Representations
in
Characteristi
p
3
Representations
and
Isogenies
4
ReductioncofcthecRingcofcEndomorphisms
5
The
Deuring
Lifting
Theorem
Chapter
14
Ihara's
Theory
1.
Deuring
Representatives
2
The
Generic
Situation
3
Special
Situations
PART
THREE
ELLIPTIC
CURVEScWITH
NON-INTEGRAL
INVARIANT
Chapter
15
The
Tate
Parametrization
1
Elliptic
Curves
with
Non-integral
Invariants
2
Ellipti
Curves Over
a
Complete
Local
Ring
Chapter
16
The
Isogeny
Theorems
1
The
Galois
p-adic
Representations
2
Results
of
Kummer
Theory
3
The
Local
Isogeny
Theorems
4
Supersingular
Redu
tion
5
The
Global
Isogeny
Theorems
Chapter
17
Division
Points
Over
Number
Fields
1
AcTheorem
of
Shafarevic
2
The
Irreducibility
Theorem
3
The
Horizontal
Galois
Group
4
The
Vertical
Galois
Group
5
End
of
the
Proof
PARTcFOURccTHETAcFUNCTIONScANDcKRONECKERcLIMIT
FORMULA
Chapter
18
Product
Expansions
1
The
Sigma
and
Zeta
Function
Appendix
The
Skew
Symmetric
Pairing
2
A
Normalization
and
the
q-product
for
the
a-function
3
q-expansions
Again
4
The
q-product
forcA
5
The
Eta
Function
of
Dedekind
6
Modular
Functions
of
Levelc2
Chapter
19
The
Siegel
Functions
and
Klein
Forms
1
The
Klein
Forms
2
The
Siegel
Functions
3
Special
Values
of
the
Siegel
Functions
Chapter
20
The
Kronecker
Limit
Formulas
1
The
Poisson
Summation
Formula
2
Examples
3
The
FunctioncKs(x)
4
The
Kronecker
First
Limit
Formula
5
The
Kronecker
Second
LimitcFormula
Chapter
21
The
First
Limit
Formula
and
L-series
1
Relation
with
L-series
2
The
Frobenius
Determinant
3
Application
to
thecL-series
Chapter
22
The
Second
Limit
Formula
and
L-series
1
Gauss
Sums
2
An
Expression
for
the
L-series
APPENDICES
ELLIPTIC
CURVES
IN
CHARACTERISTIC
p
Appendixc1
Algebraic
Formulas
in
Arbitrary
Chara
teristic
BYcJ.cTATE
1
Generalized
Weierstrass
Form
2
Canonical
Forms
3
Expansion
Near
O;
The
Formal
Group
Appendix
2
The
Tracecof
Frobenius
and
the
Differential
of
FirstcKind
1
The
Trace
of
Frobenius
2
Duality
3
The
Tate
Trace
4
The
Cartier
Operator
5
The
Hasse
Invariant
Bibliography
Index

图书封面

图书标签Tags

GTM,Springer,数学


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