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Complex multiplication: Overview, Kronecker and abelian extensions & Sample consequence

In mathematics, complex multiplication (CM) is the theory of elliptic curves E that have an endomorphism ring larger than the integers. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible when the period lattice is the Gaussian integer lattice or Eisenstein integer lattice.

Language: English [EN]
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Complex multiplication topic overview

The analysis highlights Overview, Kronecker and abelian extensions and Sample consequence as prominent areas in the source structure around Complex multiplication.

Related topics
61
Source areas
6
Connected nodes
67
Extracted relationships
38
Concept neighborhoods
44
Bridge connections
67

What this topic covers Research coverage

Source areas are shown by the number of related topics found in each part of the analysis. Use smaller areas too: they can reveal specialized angles and content gaps.

Overview · 21 topics
Kronecker and abelian extensions · 10 topics
Sample consequence · 9 topics
Singular moduli · 8 topics
Example of the imaginary quadratic field extension · 7 topics
Abstract theory of endomorphisms · 6 topics

Smaller areas are not necessarily less important. They contain fewer connections in this analysis and can be useful for finding specialized angles or coverage gaps.

Explore all related topics Closing gaps

Browse the complete topic structure, not only the most central items. Less prominent entities and concepts can reveal missing angles, specialized context and useful research gaps. Each item opens a new analysis centered on that subject.

Overview

Example of the imaginary quadratic field extension

Abstract theory of endomorphisms

Kronecker and abelian extensions

Sample consequence

Singular moduli

Advanced semantic analysis

Deeper signals for content research, entity SEO and topical coverage. The plain-language headings explain what each technical view is useful for.

How Complex multiplication connects Entity context

The extracted context around Complex multiplication shows recurring relationship patterns in the source. For example, Complex multiplication → American Mathematical Society, Bulletin, Cite, CiteSeerX, Complex, Galois Representations, Kenneth, Modular Forms, October, PlanetMath, S2CID Another extracted example is Complex multiplication → Eisenstein, Gauss, Hilbert's, Indeed, Kronecker, Kronecker Jugendtraum, Let, Shimura's, Then, This, Weierstrass. Use these groups to spot repeated connection types before inspecting the individual relationships.

Complex multiplication

Top relations

related to External links · 11
Complex multiplication → American Mathematical Society, Bulletin, Cite, CiteSeerX, Complex, Galois Representations, Kenneth, Modular Forms, October, PlanetMath, S2CID
related to Kronecker and abelian extensions · 11
Complex multiplication → Eisenstein, Gauss, Hilbert's, Indeed, Kronecker, Kronecker Jugendtraum, Let, Shimura's, Then, This, Weierstrass
related to Example of the imaginary quadratic field extension · 7
Complex multiplication → An, Consider, Conversely, Hilbert's, Kronecker, Kronecker Jugendtraum, To
related to Abstract theory of endomorphisms · 6
Complex multiplication → But, Frobenius, Hodge, It, The, When
is a · 2
Complex multiplication → exception, hardest to resolve for the Hodge conjecture
related to Singular moduli · 1
Complex multiplication → The

Important terminology

Use these terms to understand the vocabulary surrounding the topic, not as a checklist for keyword stuffing.

Important terminology

complex elliptic multiplication theory field number quadratic abelian displaystyle imaginary integers curve ring algebraic curves endomorphisms integer functions also kronecker

Complex multiplication relationships Subject–Predicate–Object triples

TTTA extracted 38 structured relationships around Complex multiplication. Examples in this analysis include Complex multiplication → is a → exception and Complex multiplication → is a → hardest to resolve for the Hodge conjecture. The table shows each extracted connection, where it came from and its confidence.

SubjectPredicateObjectConfidenceSrc
Complex multiplicationis aexception0.90text
Complex multiplicationis ahardest to resolve for the Hodge conjecture0.90text
Complex multiplicationrelated to Abstract theory of endomorphismsThe0.60section
Complex multiplicationrelated to Abstract theory of endomorphismsWhen0.60section
Complex multiplicationrelated to Abstract theory of endomorphismsFrobenius0.60section
Complex multiplicationrelated to Abstract theory of endomorphismsBut0.60section
Complex multiplicationrelated to Abstract theory of endomorphismsIt0.60section
Complex multiplicationrelated to Abstract theory of endomorphismsHodge0.60section
Complex multiplicationrelated to Example of the imaginary quadratic field extensionConsider0.60section
Complex multiplicationrelated to Example of the imaginary quadratic field extensionAn0.60section
Complex multiplicationrelated to Example of the imaginary quadratic field extensionConversely0.60section
Complex multiplicationrelated to Example of the imaginary quadratic field extensionKronecker0.60section

Related concept clusters Concept neighborhoods

The concept neighborhoods around Complex multiplication bring nearby vocabulary together. In this analysis, examples include Multiplication, Elliptic and Curve. Use the clusters to find adjacent concepts and terminology that may deserve separate research.

  • Complex multiplication
    • Multiplication
    • Elliptic
    • Curve
    • Displaystyle
    • Theory
    • Curves
    • Lambda
    • Group
    • Algebraic
    • Abelian
    • Integers
    • Ring
  • complex multiplication
    • Multiplication
    • Elliptic
    • Curve
    • Displaystyle
    • Theory
    • Curves
    • Lambda
    • Algebraic
    • Ring
    • Group
    • Field
    • Number
  • elliptic curves
    • Multiplication
    • Curve
    • Mathematics
    • Curves
    • Elliptic
    • Displaystyle
    • Functions
    • Integers
    • Ring
    • Theory
    • Lambda
    • Points
  • elliptic functions
    • Extra
    • Multiplication
    • Curve
    • Eisenstein
    • Points
    • Values
    • Curves
    • Displaystyle
    • Functions
    • Abelian
    • Integers
    • Ring
  • eisenstein integer
    • Example
    • Functions
    • Number
    • Extra
    • Extensions
    • Fields
    • Gaussian
    • Period
    • Eisenstein
    • Integer
    • Kronecker
    • Lattice
  • abelian functions
    • Kronecker
    • Extra
    • Extensions
    • Eisenstein
    • Extension
    • Points
    • Theory
    • Values
    • Also
    • Abelian
    • Functions
    • Endomorphisms
  • several complex variables
    • Multiplication
    • Elliptic
    • Curve
    • Displaystyle
    • Theory
    • Curves
    • Lambda
    • Group
    • Algebraic
    • Abelian
    • Integers
    • Ring
  • algebraic number theory
    • Function
    • Also
    • Field
    • Fields
    • Class
    • Lambda
    • Modular
    • Number
    • Group
    • Example
    • Extensions
    • Kronecker

Connections between topic areas Semantic bridges

For Complex multiplication, one of the stronger structural bridges in this analysis connects Complex multiplication with Overview. Bridges highlight paths between different parts of the map and can reveal research angles that are easy to miss in a flat list.

Min side: 3
Complex multiplication — Overview · splits 46 ⟂ 22
Complex multiplication — Kronecker and abelian extensions · splits 57 ⟂ 11
Complex multiplication — Sample consequence · splits 58 ⟂ 10
Complex multiplication — Singular moduli · splits 59 ⟂ 9
Complex multiplication — Example of the imaginary quadratic field extension · splits 60 ⟂ 8
Complex multiplication — Abstract theory of endomorphisms · splits 61 ⟂ 7

Map overview Semantic statistics

Complex multiplication

Nodes68
Edges67
Triples38
Avg. degree1.97
Density0.029412
Components1

Source & methodology

TTTA analyzes the structure around Complex multiplication to surface related topics, entities, relationships, concept neighborhoods and bridge connections. Use the map to explore areas such as Overview, Kronecker and abelian extensions & Sample consequence, including less central topics that may reveal useful research gaps. Automatically extracted connections are research leads rather than rewritten encyclopedia content.

Source: Wikipedia — Complex multiplication · EN edition · Analysis: TopicsToTalkAbout

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