Fontaine–Mazur conjecture

E885244

conjecture in number theory mathematical conjecture

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.

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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 NERFINISHED ⓘ 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 NERFINISHED ⓘ Christophe Breuil NERFINISHED ⓘ Laurent Clozel NERFINISHED ⓘ Michael Harris NERFINISHED ⓘ Richard Taylor NERFINISHED ⓘ
hasPartialResultsIn	representations attached to modular forms ⓘ two-dimensional p-adic Galois representations ⓘ
hasVariant	Fontaine–Mazur conjecture for geometric p-adic representations NERFINISHED ⓘ
implies	finiteness of certain Galois representations ⓘ
importance	central problem in modern number theory ⓘ
motivatedBy	Langlands reciprocity philosophy NERFINISHED ⓘ classification of Galois representations arising from geometry ⓘ
namedAfter	Barry Mazur NERFINISHED ⓘ Jean-Marc Fontaine NERFINISHED ⓘ
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 NERFINISHED ⓘ Taniyama–Shimura–Weil conjecture NERFINISHED ⓘ 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 NERFINISHED ⓘ
yearProposedApprox	1990s ⓘ

Referenced by (1)

Full triples — surface form annotated when it differs from this entity's canonical label.

Serre’s conjecture on Galois representations → isRelatedTo → Fontaine–Mazur conjecture ⓘ