1
Introduction
2
Background
▶
2.0.1
Arithmetic function
2.0.2
Bernoulli numbers
2.0.3
Divisor function
2.0.4
Multiplicative arithmetic function
2.0.5
Abelian variety
2.0.6
Affine space
2.0.7
Base change
2.0.8
Base field
2.0.9
Complex multiplication
2.0.10
Algebraic curve
2.0.11
Genus of a smooth curve
2.0.12
Smoothness of an algebraic curve
2.0.13
Dimension of an algebraic variety
2.0.14
Endomorphism algebra
2.0.15
Endomorphism ring
2.0.16
Geometric endomorphism ring
2.0.17
Geometrically simple
2.0.18
Hyperelliptic curve
2.0.19
Irreducible variety
2.0.20
Jacobian of a curve
2.0.21
Minimal field of definition
2.0.22
Mordell-Weil group of an abelian variety
2.0.23
Projective space
2.0.24
Quotient curve
2.0.25
Riemann surface
2.0.26
Simple
2.0.27
Non-singular point (definition)
2.0.28
Algebraic variety
2.0.29
Binary operation
2.0.30
Associative binary operation
2.0.31
Commutative binary operation
2.0.32
Identity for a binary operation
2.0.33
Inverse for a binary operation
2.0.34
Symplectic isomorphism
2.0.35
Artin representation (definition)
2.0.36
Conductor of an Artin representation
2.0.37
Number field associated to an Artin representation
2.0.38
Parity of a representation
2.0.39
Ramified prime of an Artin representation
2.0.40
Unramified prime of an Artin representation
2.0.41
Isogeny of abelian varieties
2.0.42
Simple abelian variety
2.0.43
Tate module of an abelian variety
2.0.44
Twist of an abelian variety
2.0.45
Dirichlet character
2.0.46
Conductor of a Dirichlet character
2.0.47
Galois orbit of a Dirichlet character
2.0.48
Orbit index of a Dirichlet character
2.0.49
Label of a Galois orbit of a Dirichlet character
2.0.50
Induced Dirichlet character
2.0.51
Minimal Dirichlet character
2.0.52
Modulus of a Dirichlet character
2.0.53
Order of a Dirichlet character
2.0.54
Primitive Dirichlet character
2.0.55
Principal Dirichlet character
2.0.56
Field of values of a Dirichlet character
2.0.57
Automorphism group of an algebraic curve
2.0.58
Discriminant of a genus 2 curve
2.0.59
Genus 2 curve
2.0.60
Primes of good reduction
2.0.61
Minimal equation of a hyperelliptic curve
2.0.62
Galois group
2.0.63
Borel subgroup
2.0.64
Cartan subgroup
2.0.65
Exceptional subgroup
2.0.66
Index of an open subgroup
2.0.67
Level of an open subgroup
2.0.68
Non-split Cartan subgroup
2.0.69
Normalizer of a Cartan subgroup
2.0.70
Normalizer of a non-split Cartan subgroup
2.0.71
Normalizer of a split Cartan subgroup
2.0.72
Open subgroup
2.0.73
Profinite group
2.0.74
Split Cartan subgroup
2.0.75
Definition of group
2.0.76
Abelian group
2.0.77
Automorphisms of a group
2.0.78
Characteristic subgroup
2.0.79
Coset of a subgroup
2.0.80
Frattini subgroup of a group
2.0.81
Cusps of a subgroup of the modular group
2.0.82
Width of a cusp
2.0.83
Fundamental domain
2.0.84
Absolute Galois group
2.0.85
Generators of a group
2.0.86
Haar measure of a topological group
2.0.87
Group homomorphism
2.0.88
Group isomorphism
2.0.89
Maximal subgroup of a group
2.0.90
Normal series of a group
2.0.91
Order of a group
2.0.92
Presentation of a finite group
2.0.93
Rank
2.0.94
Modular group \(\mathrm{SL}(2,\mathbb Z)\)
2.0.95
Subgroup of a group
2.0.96
Index of a subgroup
2.0.97
Normal subgroup of a group
2.0.98
Sylow subgroup
2.0.99
Torsion group
2.0.100
Automorphism group of a field extension
2.0.101
Inertia group
2.0.102
Local field
2.0.103
Maximal ideal of a local field
2.0.104
\(p\)-adic field
2.0.105
Residue field
2.0.106
Ring of integers of a local field
2.0.107
Wild inertia group
2.0.108
L-function
2.0.109
Analytic rank
2.0.110
Arithmetic L-function
2.0.111
Central point of an L-function
2.0.112
Critical line of an L-function
2.0.113
Dirichlet series
2.0.114
Dual of an L-function
2.0.115
Euler product of an L-function
2.0.116
Functional equation of an L-function
2.0.117
Gamma factors
2.0.118
Leading coefficient
2.0.119
Normalization of an L-function
2.0.120
Generalized Riemann hypothesis
2.0.121
Self-dual L-function
2.0.122
Sign of the functional equation
2.0.123
Dedekind eta function
2.0.124
Upper half-plane
2.0.125
Modular curve
2.0.126
Cusps of a modular curve
2.0.127
Level structure of a modular curve
2.0.128
Modular curve \(X(N)\)
2.0.129
Definition of ring
2.0.130
\(A\)-field
2.0.131
Characteristic of a ring
2.0.132
Dedekind domain
2.0.133
Field
2.0.134
Field of fractions of an integral domain
2.0.135
Fractional ideal
2.0.136
Ideal of a ring
2.0.137
Integral element of a ring
2.0.138
Integral domain
2.0.139
Integrally closed
2.0.140
Irreducible element
2.0.141
Maximal ideal
2.0.142
Noetherian ring
2.0.143
Prime ideal
2.0.144
Principal fractional ideal
2.0.145
Unit in a ring
2.0.146
Zero divisor
2.0.147
Euler gamma function
2.0.148
Sato-Tate group
2.0.149
Symplectic form
2.0.150
Unitary symplectic group
3
Number fields
▶
3.1
Definitions relating to number fields
▶
3.1.1
Number field
3.1.2
Abelian number field
3.1.3
Absolute discriminant of a number field
3.1.4
Absolute value of a field
3.1.5
Arithmetically equivalent fields
3.1.6
Class number of a number field
3.1.7
Analytic class number formula
3.1.8
CM number field
3.1.9
Complex embedding
3.1.10
Conductor of an abelian number field
3.1.11
Defining Polynomial of a Number Field
3.1.12
Degree of a number field
3.1.13
Dirichlet group of an Abelian number field
3.1.14
Discriminant of a number field
3.1.15
Discriminant root field
3.1.16
Embedding of a number field
3.1.17
Frobenius cycle types
3.1.18
Fundamental units of a number field
3.1.19
Galois closure of an extension
3.1.20
Galois group
3.1.21
Galois root discriminant
3.1.22
Generator of a number field
3.1.23
Ideal class group of a number field
3.1.24
Ideal labels
3.1.25
Inessential prime
3.1.26
Integral elements
3.1.27
Integral basis of a number field
3.1.28
Intermediate fields
3.1.29
Is a Galois extension
3.1.30
Local algebra
3.1.31
Maximal CM subfield
3.1.32
Minimal polynomial
3.1.33
Minimal sibling
3.1.34
Monogenic field
3.1.35
Monomial order
3.1.36
Narrow class group
3.1.37
Narrow class number
3.1.38
Number field nicknames
3.1.39
Order
3.1.40
\(p\)-adic completion of a number field
3.1.41
Place of a number field
3.1.42
Canonical defining polynomial for number fields
3.1.43
Discriminant of polynomial
3.1.44
Prime of a number field
3.1.45
Ramified (rational) prime of a number field
3.1.46
Rank of a number field
3.1.47
Real embedding
3.1.48
Reflex field
3.1.49
Reflex field of the reflex field
3.1.50
Regulator of a number field
3.1.51
Relative class number of a CM field
3.1.52
Ring of integers of a number field
3.1.53
Root discriminant of a number field
3.1.54
Separable extension
3.1.55
Separable algebra
3.1.56
Serre Odlyzko bound
3.1.57
Sibling fields and algebras
3.1.58
Signature of a number field
3.1.59
Stem field for a Galois extension
3.1.60
Unit group torsion
3.1.61
Totally imaginary
3.1.62
Totally positive
3.1.63
Totally real
3.1.64
Unit group of a number field
3.1.65
Unramified (rational) prime of a number field
3.1.66
Weil height
3.1.67
Weil polynomial
3.1.68
Index of a number field
4
Elliptic curves
▶
4.1
Definitions relating to elliptic curves over general number fields
▶
4.1.1
Elliptic curve over a field
4.1.2
Additive reduction
4.1.3
Analytic order of \(\mathop{\mathrm{\unicode {x0428}}}\nolimits \)
4.1.4
Bad reduction of an elliptic curve at a prime
4.1.5
Base change
4.1.6
Birch Swinnerton-Dyer conjecture
4.1.7
Canonical height on an elliptic curve
4.1.8
Complex multiplication
4.1.9
Conductor of an elliptic curve
4.1.10
Discriminant of a Weierstrass equation
4.1.11
Endomorphism of an elliptic curve
4.1.12
Endomorphism ring of an elliptic curve
4.1.13
Galois representations attached to an elliptic curve
4.1.14
Image of the adelic Galois representation
4.1.15
Image of mod-\(l\) Galois representation
4.1.16
Geometric endomorphism ring
4.1.17
Global minimal model
4.1.18
Good ordinary reduction
4.1.19
Good reduction
4.1.20
Good supersingular reduction
4.1.21
Integral model
4.1.22
Elliptic curve invariants
4.1.23
Isogeny between elliptic curves
4.1.24
Isogeny class of an elliptic curve
4.1.25
Isogeny class degree
4.1.26
Isogeny graph of an isogeny class of elliptic curves
4.1.27
Isogeny matrix of an isogeny class of elliptic curves
4.1.28
Isomorphism of elliptic curves
4.1.29
j-invariant of an elliptic curve
4.1.30
Kodaira symbol
4.1.31
Local data of an elliptic curve
4.1.32
Local minimal discriminant of an elliptic curve
4.1.33
Local minimal model
4.1.34
Maximal \(l\)-adic Galois representation
4.1.35
Maximal Galois representation
4.1.36
Minimal discriminant
4.1.37
Mordell-Weil group
4.1.38
Mordell-Weil theorem
4.1.39
Multiplicative reduction
4.1.40
Mordell-Weil generators
4.1.41
Non-split multiplicative reduction
4.1.42
Obstruction class of an elliptic curve
4.1.43
Tate module of an elliptic curve
4.1.44
Global period of an elliptic curve
4.1.45
Potential good reduction
4.1.46
Elliptic curve over \(\mathbb Q\)
4.1.47
\(abc\) quality
4.1.48
Analytic rank of an elliptic curve over \(\mathbb {Q}\)
4.1.49
Analytic order of \(\mathop{\mathrm{\unicode {x0428}}}\nolimits \)
4.1.50
Birch and Swinnerton-Dyer conjecture
4.1.51
Canonical height
4.1.52
Conductor of an elliptic curve over \(\mathbb {Q}\)
4.1.53
Cremona label
4.1.54
Discriminant of an elliptic curve over \(\mathbb {Q}\)
4.1.55
Endomorphism ring of an elliptic curve
4.1.56
Faltings height of an elliptic curve
4.1.57
Faltings ratio
4.1.58
Frey curve
4.1.59
Integral points
4.1.60
j-invariant of a rational elliptic curve
4.1.61
Kodaira symbol
4.1.62
Label for an elliptic curve over \(\mathbb Q\)
4.1.63
Manin constant for elliptic curves over \(\mathbb {Q}\)
4.1.64
Minimal twists of elliptic curves over \(\mathbb {Q}\)
4.1.65
Minimal Weierstrass equation over \(\mathbb Q\)
4.1.66
Modular degree of an elliptic curve over \(\mathbb Q\)
4.1.67
Modular parametrization of an elliptic curve over \(\mathbb Q\)
4.1.68
Naive height
4.1.69
Optimal elliptic curve over \(\mathbb Q\)
4.1.70
Period lattice of an elliptic curve
4.1.71
Real period
4.1.72
Reduction type of an elliptic curve over \(\mathbb Q\)
4.1.73
Regulator of elliptic curve
4.1.74
Semistable elliptic curve
4.1.75
Serre invariants
4.1.76
Special value of an elliptic curve L-function
4.1.77
Szpiro ratio
4.1.78
Torsion growth in number fields
4.1.79
Torsion subgroup of an elliptic curve over \(\mathbb Q\)
4.1.80
\(\mathbb {Q}\)-curves
4.1.81
Rank of an elliptic curve over a number field
4.1.82
Reduction of an elliptic curve
4.1.83
Reduction type
4.1.84
Regulator of an elliptic curve
4.1.85
Elliptic curve over a ring
4.1.86
Elliptic scheme
4.1.87
Semi-global minimal model
4.1.88
Semistable elliptic curve
4.1.89
Simplified equation
4.1.90
Special value of an elliptic curve L-function
4.1.91
Split multiplicative reduction
4.1.92
Tamagawa number
4.1.93
Torsion order of an elliptic curve
4.1.94
Torsion subgroup of an elliptic curve
4.1.95
Twists of elliptic curves
4.1.96
Weierstrass equation or model
4.1.97
Isomorphism between Weierstrass models
5
Modular forms
▶
5.1
Definitions relating to classical modular forms
▶
5.1.1
Classical modular form
5.1.2
Analytic conductor of a classical newform
5.1.3
Analytic rank
5.1.4
Artin field
5.1.5
Artin image
5.1.6
Atkin-Lehner involution \(w_Q\)
5.1.7
Bad prime
5.1.8
Character of a modular form
5.1.9
CM form
5.1.10
Coefficient field for newforms
5.1.11
Coefficient ring
5.1.12
Congruence subgroup
5.1.13
Cuspidal modular form
5.1.14
Decomposition into newforms
5.1.15
Defining polynomial
5.1.16
Dimension
5.1.17
Distinguishing Hecke operators
5.1.18
Dual cuspform
5.1.19
Holomorphic Eisenstein series of level 1
5.1.20
Holomorphic Eisenstein modular form
5.1.21
Label of a classical Eisenstein modular form
5.1.22
Eisenstein newform
5.1.23
New Eisenstein subspace
5.1.24
Holomorphic Eisenstein series
5.1.25
Embedding of a modular form
5.1.26
Complex embedding label
5.1.27
Eta quotient
5.1.28
Fourier coefficients of a modular form
5.1.29
Fricke involution
5.1.30
Galois conjugate newforms
5.1.31
Galois orbit of a newform
5.1.32
Galois representation
5.1.33
Hecke operator
5.1.34
Hecke orbit
5.1.35
Coefficient ring generator bound
5.1.36
Hecke characteristic polynomial
5.1.37
Inner twist
5.1.38
Inner twist count
5.1.39
Inner twist multiplicity
5.1.40
Label of a classical modular form
5.1.41
Label of a classical modular form
5.1.42
Level of a modular form
5.1.43
Maximal newform
5.1.44
Minimal modular form
5.1.45
Minimal twist
5.1.46
Minus space
5.1.47
Newform
5.1.48
Newform subspace
5.1.49
New subspace
5.1.50
Nontrivial inner twist
5.1.51
Old subspace of modular forms
5.1.52
Petersson scalar product
5.1.53
Plus space
5.1.54
Projective field
5.1.55
Projective image
5.1.56
q-expansion of a modular form
5.1.57
Relative dimension
5.1.58
Real multiplication
5.1.59
Satake Angles
5.1.60
Satake parameters
5.1.61
Sato-Tate group of a modular form
5.1.62
Self-twist
5.1.63
Self dual modular form
5.1.64
Shimura correspondence
5.1.65
Spaces of modular forms
5.1.66
Trace form
5.1.67
Stark unit of a newform of weight one
5.1.68
Sturm bound
5.1.69
Sturm bound for Gamma1(N)
5.1.70
Subspaces of modular forms
5.1.71
Trace bound
5.1.72
Trace form
5.1.73
Twist
5.1.74
Twist minimal
5.1.75
Twist multiplicity
5.1.76
Weight of an elliptic modular form
Dependency graph
Lean–LMFDB bridge
Chris Birkbeck David Roe Andrew Sutherland
1
Introduction
2
Background
2.0.1
Arithmetic function
2.0.2
Bernoulli numbers
2.0.3
Divisor function
2.0.4
Multiplicative arithmetic function
2.0.5
Abelian variety
2.0.6
Affine space
2.0.7
Base change
2.0.8
Base field
2.0.9
Complex multiplication
2.0.10
Algebraic curve
2.0.11
Genus of a smooth curve
2.0.12
Smoothness of an algebraic curve
2.0.13
Dimension of an algebraic variety
2.0.14
Endomorphism algebra
2.0.15
Endomorphism ring
2.0.16
Geometric endomorphism ring
2.0.17
Geometrically simple
2.0.18
Hyperelliptic curve
2.0.19
Irreducible variety
2.0.20
Jacobian of a curve
2.0.21
Minimal field of definition
2.0.22
Mordell-Weil group of an abelian variety
2.0.23
Projective space
2.0.24
Quotient curve
2.0.25
Riemann surface
2.0.26
Simple
2.0.27
Non-singular point (definition)
2.0.28
Algebraic variety
2.0.29
Binary operation
2.0.30
Associative binary operation
2.0.31
Commutative binary operation
2.0.32
Identity for a binary operation
2.0.33
Inverse for a binary operation
2.0.34
Symplectic isomorphism
2.0.35
Artin representation (definition)
2.0.36
Conductor of an Artin representation
2.0.37
Number field associated to an Artin representation
2.0.38
Parity of a representation
2.0.39
Ramified prime of an Artin representation
2.0.40
Unramified prime of an Artin representation
2.0.41
Isogeny of abelian varieties
2.0.42
Simple abelian variety
2.0.43
Tate module of an abelian variety
2.0.44
Twist of an abelian variety
2.0.45
Dirichlet character
2.0.46
Conductor of a Dirichlet character
2.0.47
Galois orbit of a Dirichlet character
2.0.48
Orbit index of a Dirichlet character
2.0.49
Label of a Galois orbit of a Dirichlet character
2.0.50
Induced Dirichlet character
2.0.51
Minimal Dirichlet character
2.0.52
Modulus of a Dirichlet character
2.0.53
Order of a Dirichlet character
2.0.54
Primitive Dirichlet character
2.0.55
Principal Dirichlet character
2.0.56
Field of values of a Dirichlet character
2.0.57
Automorphism group of an algebraic curve
2.0.58
Discriminant of a genus 2 curve
2.0.59
Genus 2 curve
2.0.60
Primes of good reduction
2.0.61
Minimal equation of a hyperelliptic curve
2.0.62
Galois group
2.0.63
Borel subgroup
2.0.64
Cartan subgroup
2.0.65
Exceptional subgroup
2.0.66
Index of an open subgroup
2.0.67
Level of an open subgroup
2.0.68
Non-split Cartan subgroup
2.0.69
Normalizer of a Cartan subgroup
2.0.70
Normalizer of a non-split Cartan subgroup
2.0.71
Normalizer of a split Cartan subgroup
2.0.72
Open subgroup
2.0.73
Profinite group
2.0.74
Split Cartan subgroup
2.0.75
Definition of group
2.0.76
Abelian group
2.0.77
Automorphisms of a group
2.0.78
Characteristic subgroup
2.0.79
Coset of a subgroup
2.0.80
Frattini subgroup of a group
2.0.81
Cusps of a subgroup of the modular group
2.0.82
Width of a cusp
2.0.83
Fundamental domain
2.0.84
Absolute Galois group
2.0.85
Generators of a group
2.0.86
Haar measure of a topological group
2.0.87
Group homomorphism
2.0.88
Group isomorphism
2.0.89
Maximal subgroup of a group
2.0.90
Normal series of a group
2.0.91
Order of a group
2.0.92
Presentation of a finite group
2.0.93
Rank
2.0.94
Modular group \(\mathrm{SL}(2,\mathbb Z)\)
2.0.95
Subgroup of a group
2.0.96
Index of a subgroup
2.0.97
Normal subgroup of a group
2.0.98
Sylow subgroup
2.0.99
Torsion group
2.0.100
Automorphism group of a field extension
2.0.101
Inertia group
2.0.102
Local field
2.0.103
Maximal ideal of a local field
2.0.104
\(p\)-adic field
2.0.105
Residue field
2.0.106
Ring of integers of a local field
2.0.107
Wild inertia group
2.0.108
L-function
2.0.109
Analytic rank
2.0.110
Arithmetic L-function
2.0.111
Central point of an L-function
2.0.112
Critical line of an L-function
2.0.113
Dirichlet series
2.0.114
Dual of an L-function
2.0.115
Euler product of an L-function
2.0.116
Functional equation of an L-function
2.0.117
Gamma factors
2.0.118
Leading coefficient
2.0.119
Normalization of an L-function
2.0.120
Generalized Riemann hypothesis
2.0.121
Self-dual L-function
2.0.122
Sign of the functional equation
2.0.123
Dedekind eta function
2.0.124
Upper half-plane
2.0.125
Modular curve
2.0.126
Cusps of a modular curve
2.0.127
Level structure of a modular curve
2.0.128
Modular curve \(X(N)\)
2.0.129
Definition of ring
2.0.130
\(A\)-field
2.0.131
Characteristic of a ring
2.0.132
Dedekind domain
2.0.133
Field
2.0.134
Field of fractions of an integral domain
2.0.135
Fractional ideal
2.0.136
Ideal of a ring
2.0.137
Integral element of a ring
2.0.138
Integral domain
2.0.139
Integrally closed
2.0.140
Irreducible element
2.0.141
Maximal ideal
2.0.142
Noetherian ring
2.0.143
Prime ideal
2.0.144
Principal fractional ideal
2.0.145
Unit in a ring
2.0.146
Zero divisor
2.0.147
Euler gamma function
2.0.148
Sato-Tate group
2.0.149
Symplectic form
2.0.150
Unitary symplectic group
3
Number fields
3.1
Definitions relating to number fields
3.1.1
Number field
3.1.2
Abelian number field
3.1.3
Absolute discriminant of a number field
3.1.4
Absolute value of a field
3.1.5
Arithmetically equivalent fields
3.1.6
Class number of a number field
3.1.7
Analytic class number formula
3.1.8
CM number field
3.1.9
Complex embedding
3.1.10
Conductor of an abelian number field
3.1.11
Defining Polynomial of a Number Field
3.1.12
Degree of a number field
3.1.13
Dirichlet group of an Abelian number field
3.1.14
Discriminant of a number field
3.1.15
Discriminant root field
3.1.16
Embedding of a number field
3.1.17
Frobenius cycle types
3.1.18
Fundamental units of a number field
3.1.19
Galois closure of an extension
3.1.20
Galois group
3.1.21
Galois root discriminant
3.1.22
Generator of a number field
3.1.23
Ideal class group of a number field
3.1.24
Ideal labels
3.1.25
Inessential prime
3.1.26
Integral elements
3.1.27
Integral basis of a number field
3.1.28
Intermediate fields
3.1.29
Is a Galois extension
3.1.30
Local algebra
3.1.31
Maximal CM subfield
3.1.32
Minimal polynomial
3.1.33
Minimal sibling
3.1.34
Monogenic field
3.1.35
Monomial order
3.1.36
Narrow class group
3.1.37
Narrow class number
3.1.38
Number field nicknames
3.1.39
Order
3.1.40
\(p\)-adic completion of a number field
3.1.41
Place of a number field
3.1.42
Canonical defining polynomial for number fields
3.1.43
Discriminant of polynomial
3.1.44
Prime of a number field
3.1.45
Ramified (rational) prime of a number field
3.1.46
Rank of a number field
3.1.47
Real embedding
3.1.48
Reflex field
3.1.49
Reflex field of the reflex field
3.1.50
Regulator of a number field
3.1.51
Relative class number of a CM field
3.1.52
Ring of integers of a number field
3.1.53
Root discriminant of a number field
3.1.54
Separable extension
3.1.55
Separable algebra
3.1.56
Serre Odlyzko bound
3.1.57
Sibling fields and algebras
3.1.58
Signature of a number field
3.1.59
Stem field for a Galois extension
3.1.60
Unit group torsion
3.1.61
Totally imaginary
3.1.62
Totally positive
3.1.63
Totally real
3.1.64
Unit group of a number field
3.1.65
Unramified (rational) prime of a number field
3.1.66
Weil height
3.1.67
Weil polynomial
3.1.68
Index of a number field
4
Elliptic curves
4.1
Definitions relating to elliptic curves over general number fields
4.1.1
Elliptic curve over a field
4.1.2
Additive reduction
4.1.3
Analytic order of \(\mathop{\mathrm{\unicode {x0428}}}\nolimits \)
4.1.4
Bad reduction of an elliptic curve at a prime
4.1.5
Base change
4.1.6
Birch Swinnerton-Dyer conjecture
4.1.7
Canonical height on an elliptic curve
4.1.8
Complex multiplication
4.1.9
Conductor of an elliptic curve
4.1.10
Discriminant of a Weierstrass equation
4.1.11
Endomorphism of an elliptic curve
4.1.12
Endomorphism ring of an elliptic curve
4.1.13
Galois representations attached to an elliptic curve
4.1.14
Image of the adelic Galois representation
4.1.15
Image of mod-\(l\) Galois representation
4.1.16
Geometric endomorphism ring
4.1.17
Global minimal model
4.1.18
Good ordinary reduction
4.1.19
Good reduction
4.1.20
Good supersingular reduction
4.1.21
Integral model
4.1.22
Elliptic curve invariants
4.1.23
Isogeny between elliptic curves
4.1.24
Isogeny class of an elliptic curve
4.1.25
Isogeny class degree
4.1.26
Isogeny graph of an isogeny class of elliptic curves
4.1.27
Isogeny matrix of an isogeny class of elliptic curves
4.1.28
Isomorphism of elliptic curves
4.1.29
j-invariant of an elliptic curve
4.1.30
Kodaira symbol
4.1.31
Local data of an elliptic curve
4.1.32
Local minimal discriminant of an elliptic curve
4.1.33
Local minimal model
4.1.34
Maximal \(l\)-adic Galois representation
4.1.35
Maximal Galois representation
4.1.36
Minimal discriminant
4.1.37
Mordell-Weil group
4.1.38
Mordell-Weil theorem
4.1.39
Multiplicative reduction
4.1.40
Mordell-Weil generators
4.1.41
Non-split multiplicative reduction
4.1.42
Obstruction class of an elliptic curve
4.1.43
Tate module of an elliptic curve
4.1.44
Global period of an elliptic curve
4.1.45
Potential good reduction
4.1.46
Elliptic curve over \(\mathbb Q\)
4.1.47
\(abc\) quality
4.1.48
Analytic rank of an elliptic curve over \(\mathbb {Q}\)
4.1.49
Analytic order of \(\mathop{\mathrm{\unicode {x0428}}}\nolimits \)
4.1.50
Birch and Swinnerton-Dyer conjecture
4.1.51
Canonical height
4.1.52
Conductor of an elliptic curve over \(\mathbb {Q}\)
4.1.53
Cremona label
4.1.54
Discriminant of an elliptic curve over \(\mathbb {Q}\)
4.1.55
Endomorphism ring of an elliptic curve
4.1.56
Faltings height of an elliptic curve
4.1.57
Faltings ratio
4.1.58
Frey curve
4.1.59
Integral points
4.1.60
j-invariant of a rational elliptic curve
4.1.61
Kodaira symbol
4.1.62
Label for an elliptic curve over \(\mathbb Q\)
4.1.63
Manin constant for elliptic curves over \(\mathbb {Q}\)
4.1.64
Minimal twists of elliptic curves over \(\mathbb {Q}\)
4.1.65
Minimal Weierstrass equation over \(\mathbb Q\)
4.1.66
Modular degree of an elliptic curve over \(\mathbb Q\)
4.1.67
Modular parametrization of an elliptic curve over \(\mathbb Q\)
4.1.68
Naive height
4.1.69
Optimal elliptic curve over \(\mathbb Q\)
4.1.70
Period lattice of an elliptic curve
4.1.71
Real period
4.1.72
Reduction type of an elliptic curve over \(\mathbb Q\)
4.1.73
Regulator of elliptic curve
4.1.74
Semistable elliptic curve
4.1.75
Serre invariants
4.1.76
Special value of an elliptic curve L-function
4.1.77
Szpiro ratio
4.1.78
Torsion growth in number fields
4.1.79
Torsion subgroup of an elliptic curve over \(\mathbb Q\)
4.1.80
\(\mathbb {Q}\)-curves
4.1.81
Rank of an elliptic curve over a number field
4.1.82
Reduction of an elliptic curve
4.1.83
Reduction type
4.1.84
Regulator of an elliptic curve
4.1.85
Elliptic curve over a ring
4.1.86
Elliptic scheme
4.1.87
Semi-global minimal model
4.1.88
Semistable elliptic curve
4.1.89
Simplified equation
4.1.90
Special value of an elliptic curve L-function
4.1.91
Split multiplicative reduction
4.1.92
Tamagawa number
4.1.93
Torsion order of an elliptic curve
4.1.94
Torsion subgroup of an elliptic curve
4.1.95
Twists of elliptic curves
4.1.96
Weierstrass equation or model
4.1.97
Isomorphism between Weierstrass models
5
Modular forms
5.1
Definitions relating to classical modular forms
5.1.1
Classical modular form
5.1.2
Analytic conductor of a classical newform
5.1.3
Analytic rank
5.1.4
Artin field
5.1.5
Artin image
5.1.6
Atkin-Lehner involution \(w_Q\)
5.1.7
Bad prime
5.1.8
Character of a modular form
5.1.9
CM form
5.1.10
Coefficient field for newforms
5.1.11
Coefficient ring
5.1.12
Congruence subgroup
5.1.13
Cuspidal modular form
5.1.14
Decomposition into newforms
5.1.15
Defining polynomial
5.1.16
Dimension
5.1.17
Distinguishing Hecke operators
5.1.18
Dual cuspform
5.1.19
Holomorphic Eisenstein series of level 1
5.1.20
Holomorphic Eisenstein modular form
5.1.21
Label of a classical Eisenstein modular form
5.1.22
Eisenstein newform
5.1.23
New Eisenstein subspace
5.1.24
Holomorphic Eisenstein series
5.1.25
Embedding of a modular form
5.1.26
Complex embedding label
5.1.27
Eta quotient
5.1.28
Fourier coefficients of a modular form
5.1.29
Fricke involution
5.1.30
Galois conjugate newforms
5.1.31
Galois orbit of a newform
5.1.32
Galois representation
5.1.33
Hecke operator
5.1.34
Hecke orbit
5.1.35
Coefficient ring generator bound
5.1.36
Hecke characteristic polynomial
5.1.37
Inner twist
5.1.38
Inner twist count
5.1.39
Inner twist multiplicity
5.1.40
Label of a classical modular form
5.1.41
Label of a classical modular form
5.1.42
Level of a modular form
5.1.43
Maximal newform
5.1.44
Minimal modular form
5.1.45
Minimal twist
5.1.46
Minus space
5.1.47
Newform
5.1.48
Newform subspace
5.1.49
New subspace
5.1.50
Nontrivial inner twist
5.1.51
Old subspace of modular forms
5.1.52
Petersson scalar product
5.1.53
Plus space
5.1.54
Projective field
5.1.55
Projective image
5.1.56
q-expansion of a modular form
5.1.57
Relative dimension
5.1.58
Real multiplication
5.1.59
Satake Angles
5.1.60
Satake parameters
5.1.61
Sato-Tate group of a modular form
5.1.62
Self-twist
5.1.63
Self dual modular form
5.1.64
Shimura correspondence
5.1.65
Spaces of modular forms
5.1.66
Trace form
5.1.67
Stark unit of a newform of weight one
5.1.68
Sturm bound
5.1.69
Sturm bound for Gamma1(N)
5.1.70
Subspaces of modular forms
5.1.71
Trace bound
5.1.72
Trace form
5.1.73
Twist
5.1.74
Twist minimal
5.1.75
Twist multiplicity
5.1.76
Weight of an elliptic modular form