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Modularity theorem: History & Art

In number theory, the modularity theorem states that elliptic curves over the field of rational numbers are related to modular forms in a particular way. Andrew Wiles and Richard Taylor proved the modularity theorem for semistable elliptic curves, which was enough to imply Fermat's Last Theorem (FLT). Later, a series of papers by Wiles's former students…

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Modularity theorem topic overview

The analysis highlights History and Art as prominent areas in the source structure around Modularity theorem.

Related topics
53
Source areas
5
Connected nodes
58
Extracted relationships
16
Related term clusters
30
Bridge connections
58

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.

Statement · 15 topics
History · 14 topics
Overview · 11 topics
Example · 7 topics
Generalizations · 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.

Key facts & relationships

High-confidence facts extracted from structured source data. Use them as anchors for further research.

Conjectured by
Yutaka Taniyama Goro Shimura
Conjectured in
1957
Consequences
Fermat's Last Theorem
Field
Number theory
First proof by
Christophe Breuil Brian Conrad Fred Diamond Richard Taylor
First proof in
2001

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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

Statement

History

Generalizations

Example

For the semantics nerds

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Advanced semantic analysis

How Modularity theorem connects Entity context

The extracted context around Modularity theorem shows recurring relationship patterns in the source. For example, Modularity theorem → Freitas, Le Hung, Robert Langlands, Siksek, The Langlands Another extracted example is Modularity theorem → Dirichlet, L-series, The L-series. Use these groups to spot repeated connection types before inspecting the individual relationships.

Modularity theorem

Top relations

related to Generalizations · 5
Modularity theorem → Freitas, Le Hung, Robert Langlands, Siksek, The Langlands
related to Related statements · 3
Modularity theorem → Dirichlet, L-series, The L-series
Conjectured by · 1
Modularity theorem → Yutaka Taniyama Goro Shimura
Conjectured in · 1
Modularity theorem → 1957
Consequences · 1
Modularity theorem → Fermat's Last Theorem
Field · 1
Modularity theorem → Number theory
First proof by · 1
Modularity theorem → Christophe Breuil Brian Conrad Fred Diamond Richard Taylor
First proof in · 1
Modularity theorem → 2001
is a · 1
Modularity theorem → special case of more general conjectures due to Robert Langlands
related to Statement · 1
Modularity theorem → X0

Important terminology

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

Important terminology

elliptic theorem conjecture modular modularity curve taniyama shimura curves flt weil proof number theory taylor proved conductor richard level form

Modularity theorem relationships Subject–Predicate–Object triples

TTTA extracted 16 structured relationships around Modularity theorem. Examples in this analysis include Modularity theorem → Conjectured by → Yutaka Taniyama Goro Shimura and Modularity theorem → Conjectured in → 1957. The table shows each extracted connection, where it came from and its confidence.

SubjectPredicateObjectConfidenceSrc
Modularity theoremConjectured byYutaka Taniyama Goro Shimura1.00infobox
Modularity theoremConjectured in19571.00infobox
Modularity theoremConsequencesFermat's Last Theorem1.00infobox
Modularity theoremFieldNumber theory1.00infobox
Modularity theoremFirst proof byChristophe Breuil Brian Conrad Fred Diamond Richard Taylor1.00infobox
Modularity theoremFirst proof in20011.00infobox
Modularity theoremis aspecial case of more general conjectures due to Robert Langlands0.90text
Modularity theoremrelated to GeneralizationsRobert Langlands0.60section
Modularity theoremrelated to GeneralizationsThe Langlands0.60section
Modularity theoremrelated to GeneralizationsFreitas0.60section
Modularity theoremrelated to GeneralizationsLe Hung0.60section
Modularity theoremrelated to GeneralizationsSiksek0.60section

Related concept clusters Related term clusters

The concept neighborhoods around Modularity theorem bring nearby vocabulary together. In this analysis, examples include Theorem, Last and Proof. Use the clusters to find adjacent concepts and terminology that may deserve separate research.

  • Modularity theorem
    • Theorem
    • Last
    • Proof
    • Fermat's
    • Known
    • Elliptic
    • Number
    • Curves
    • Related
    • Statement
    • Breuil
    • Conrad
  • modularity theorem
    • Theorem
    • Last
    • Proof
    • Fermat's
    • Known
    • Related
    • Richard
    • Taylor
    • Conjecture
    • Elliptic
    • Number
    • Theory
  • number theory
    • Theory
    • Field
    • Related
    • Theorem
    • Modular
    • Rational
    • Modularity
    • Form
    • Elliptic
    • Taniyama
    • Breuil
    • Conrad
  • elliptic curves
    • Curves
    • Elliptic
    • Curve
    • Proved
    • Modular
    • Andrew
    • Wiles
    • Rational
    • Weil
    • Modularity
    • Shimura
    • Forms
  • modular forms
    • Curve
    • Defined
    • Weight
    • Level
    • Number
    • Forms
    • Modular
    • Rational
    • Conductor
    • Form
    • Related
    • Modularity
  • semistable elliptic curves
    • Curves
    • Elliptic
    • Curve
    • Proved
    • Modular
    • Andrew
    • Wiles
    • Rational
    • Weil
    • Modularity
    • Shimura
    • Forms
  • fermat's last theorem
    • Last
    • Richard
    • Proof
    • Taylor
    • Theorem
    • Wiles
    • Fermat's
    • Related
    • Flt
    • Known
    • Breuil
    • Conrad
  • brian conrad
    • Breuil
    • Diamond
    • Taylor
    • Prove
    • Richard
    • Fermat's
    • Field
    • Related
    • Statement
    • Shimura
    • Last
    • Rational

Connections between topic areas Semantic bridges

For Modularity theorem, one of the stronger structural bridges in this analysis connects Modularity theorem with Statement. 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
Modularity theorem — Statement · splits 43 ⟂ 16
Modularity theorem — History · splits 44 ⟂ 15
Modularity theorem — Overview · splits 47 ⟂ 12
Modularity theorem — Example · splits 51 ⟂ 8
Modularity theorem — Generalizations · splits 52 ⟂ 7

Map overview Semantic statistics

Modularity theorem

Nodes59
Edges58
Triples16
Avg. degree1.97
Density0.033898
Components1

Source & methodology

TTTA analyzes the structure around Modularity theorem to surface related topics, entities, relationships, concept neighborhoods and bridge connections. Use the map to explore areas such as History & Art, including less central topics that may reveal useful research gaps. Automatically extracted connections are research leads rather than rewritten encyclopedia content.

Source: Wikipedia — Modularity theorem · EN edition · Analysis: TopicsToTalkAbout

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