p-adic Hodge theory

E551974

branch of arithmetic geometry branch of mathematics

p-adic Hodge theory is a branch of arithmetic geometry that studies p-adic Galois representations and their relationship to the cohomology of algebraic varieties over p-adic fields, using analogues of classical Hodge-theoretic structures.

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Statements (50)

Predicate	Object
instanceOf	branch of arithmetic geometry ⓘ branch of mathematics ⓘ
appliesTo	Galois representations of local fields ⓘ étale cohomology of varieties over p-adic fields ⓘ
centralConcept	Hodge–Tate weights NERFINISHED ⓘ comparison isomorphisms ⓘ p-adic period isomorphisms ⓘ
developedBy	Jean-Marc Fontaine NERFINISHED ⓘ
fieldOfStudy	cohomology of algebraic varieties over p-adic fields ⓘ p-adic Galois representations ⓘ p-adic Hodge structures ⓘ
hasApplication	study of Galois representations attached to automorphic forms ⓘ study of modular forms ⓘ
hasGoal	classify p-adic Galois representations via linear algebra data ⓘ relate arithmetic invariants to geometric invariants ⓘ
hasSubfield	(φ,Γ)-module theory ⓘ integral p-adic Hodge theory ⓘ relative p-adic Hodge theory ⓘ
historicalPeriod	late 20th century ⓘ
influenced	modern arithmetic geometry ⓘ p-adic representation theory ⓘ theory of eigenvarieties ⓘ
influencedBy	Grothendieck’s theory of schemes ⓘ Hodge theory NERFINISHED ⓘ étale cohomology theory ⓘ
involvesConstruction	B_HT NERFINISHED ⓘ B_cris ⓘ B_dR NERFINISHED ⓘ B_st ⓘ Fontaine period rings NERFINISHED ⓘ
relatedTo	Iwasawa theory NERFINISHED ⓘ algebraic geometry ⓘ classical Hodge theory ⓘ motivic cohomology ⓘ number theory ⓘ p-adic Langlands program ⓘ
studiesProperty	Hodge–Tate decomposition NERFINISHED ⓘ comparison between different cohomology theories ⓘ crystalline representations ⓘ de Rham representations NERFINISHED ⓘ semistable representations ⓘ structure of p-adic Galois representations ⓘ
usesConcept	Galois representations NERFINISHED ⓘ crystalline cohomology ⓘ de Rham cohomology NERFINISHED ⓘ p-adic differential equations ⓘ p-adic fields ⓘ period rings ⓘ rigid cohomology ⓘ étale cohomology ⓘ

Referenced by (1)

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

Hodge theory → hasSubfield → p-adic Hodge theory ⓘ