Fontaine–Mazur conjecture
E885244
The Fontaine–Mazur conjecture is a central open problem in number theory that predicts which p-adic Galois representations of number fields arise from geometry or from automorphic forms.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Fontaine–Mazur conjecture canonical | 1 |
| Fontaine–Mazur conjecture for geometric p-adic representations | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10732858 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Fontaine–Mazur conjecture Context triple: [Serre’s conjecture on Galois representations, isRelatedTo, Fontaine–Mazur conjecture]
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A.
Taniyama–Shimura–Weil conjecture
The Taniyama–Shimura–Weil conjecture, now the modularity theorem, asserts that every elliptic curve over the rational numbers is modular and played a central role in the proof of Fermat’s Last Theorem.
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B.
Birch and Swinnerton-Dyer Conjecture
The Birch and Swinnerton-Dyer Conjecture is a central unsolved problem in number theory that predicts a deep connection between the arithmetic of rational points on an elliptic curve and the behavior of its associated L-function at a specific value.
-
C.
Tate Conjecture
The Tate Conjecture is a major open problem in arithmetic geometry that predicts a deep connection between algebraic cycles on varieties over finite fields and their Galois-invariant étale cohomology classes.
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D.
Serre’s conjecture on Galois representations
Serre’s conjecture on Galois representations is a landmark statement in number theory that predicts which two-dimensional mod p Galois representations of the absolute Galois group of the rationals arise from modular forms.
-
E.
Ramanujan–Petersson conjecture
The Ramanujan–Petersson conjecture is a fundamental statement in number theory and the theory of modular forms that predicts strong bounds on the Fourier coefficients of modular cusp forms, with deep connections to automorphic forms and the Langlands program.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Fontaine–Mazur conjecture Target entity description: The Fontaine–Mazur conjecture is a central open problem in number theory that predicts which p-adic Galois representations of number fields arise from geometry or from automorphic forms.
-
A.
Taniyama–Shimura–Weil conjecture
The Taniyama–Shimura–Weil conjecture, now the modularity theorem, asserts that every elliptic curve over the rational numbers is modular and played a central role in the proof of Fermat’s Last Theorem.
-
B.
Birch and Swinnerton-Dyer Conjecture
The Birch and Swinnerton-Dyer Conjecture is a central unsolved problem in number theory that predicts a deep connection between the arithmetic of rational points on an elliptic curve and the behavior of its associated L-function at a specific value.
-
C.
Tate Conjecture
The Tate Conjecture is a major open problem in arithmetic geometry that predicts a deep connection between algebraic cycles on varieties over finite fields and their Galois-invariant étale cohomology classes.
-
D.
Serre’s conjecture on Galois representations
Serre’s conjecture on Galois representations is a landmark statement in number theory that predicts which two-dimensional mod p Galois representations of the absolute Galois group of the rationals arise from modular forms.
-
E.
Ramanujan–Petersson conjecture
The Ramanujan–Petersson conjecture is a fundamental statement in number theory and the theory of modular forms that predicts strong bounds on the Fourier coefficients of modular cusp forms, with deep connections to automorphic forms and the Langlands program.
- F. None of above. chosen
Statements (43)
| Predicate | Object |
|---|---|
| instanceOf |
conjecture in number theory
ⓘ
mathematical conjecture ⓘ |
| appliesTo | continuous p-adic representations of absolute Galois groups of number fields ⓘ |
| assumes |
potentially semistable at primes above p
ⓘ
unramified outside finitely many primes ⓘ |
| concerns |
automorphic Galois representations
ⓘ
geometric Galois representations ⓘ p-adic Galois representations of number fields ⓘ |
| conclusion | such representations come from geometry or automorphic forms ⓘ |
| context |
Langlands program
ⓘ
relationship between Galois representations and automorphic forms ⓘ |
| discussedIn | research articles in arithmetic geometry and the Langlands program ⓘ |
| field | number theory ⓘ |
| formulationLanguage | p-adic Hodge-theoretic conditions ⓘ |
| hasConsequence | constraints on non-geometric p-adic Galois representations ⓘ |
| hasPartialResultsBy |
Andrew Wiles
ⓘ
Christophe Breuil ⓘ Laurent Clozel ⓘ Michael Harris ⓘ Richard Taylor ⓘ |
| hasPartialResultsIn |
representations attached to modular forms
ⓘ
two-dimensional p-adic Galois representations ⓘ |
| hasVariant |
Fontaine–Mazur conjecture for geometric p-adic representations
ⓘ
linked to:
Fontaine–Mazur conjecture
|
| implies | finiteness of certain Galois representations ⓘ |
| importance | central problem in modern number theory ⓘ |
| motivatedBy |
Langlands reciprocity philosophy
ⓘ
linked to:
Langlands program
classification of Galois representations arising from geometry ⓘ |
| namedAfter |
Barry Mazur
ⓘ
Jean-Marc Fontaine ⓘ |
| openAsOf | 2024 ⓘ |
| predicts |
which p-adic Galois representations arise from automorphic forms
ⓘ
which p-adic Galois representations arise from geometry ⓘ |
| relatedTo |
Grothendieck’s theory of motives
ⓘ
Serre conjecture ⓘ
linked to:
Serre’s conjecture on Galois representations
Taniyama–Shimura–Weil conjecture ⓘ modularity of Galois representations ⓘ p-adic Hodge theory classification of representations ⓘ |
| status | open problem ⓘ |
| subfield |
Galois representations
ⓘ
arithmetic geometry ⓘ automorphic forms ⓘ p-adic Hodge theory ⓘ |
| yearProposedApprox | 1990s ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Fontaine–Mazur conjecture Description of subject: The Fontaine–Mazur conjecture is a central open problem in number theory that predicts which p-adic Galois representations of number fields arise from geometry or from automorphic forms.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
Fontaine–Mazur conjecture
→
hasVariant
→
Fontaine–Mazur conjecture for geometric p-adic representations
ⓘ
linked to: Fontaine–Mazur conjecture