p-adic Hodge theory
E551974
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.
All labels observed (7)
| Label | Occurrences |
|---|---|
| p-adic Hodge theory canonical | 9 |
| B_{cris} (crystalline period ring) | 1 |
| B_{st} (semistable period ring) | 1 |
| Fontaine’s theory of (φ,Γ)-modules | 1 |
| Hodge–Tate decomposition | 1 |
| Hodge–Tate representation | 1 |
| Hodge–Tate weights | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5837393 — 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: p-adic Hodge theory Context triple: [Hodge theory, hasSubfield, p-adic Hodge theory]
-
A.
Galois representations
Galois representations are homomorphisms from Galois groups of field extensions into matrix groups that encode deep arithmetic information and link number theory with algebraic geometry and modular forms.
-
B.
Iwasawa theory
Iwasawa theory is a branch of number theory that studies the growth of arithmetic invariants in infinite towers of number fields, particularly using p-adic methods.
-
C.
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.
-
D.
Hasse–Arf theorem
The Hasse–Arf theorem is a fundamental result in algebraic number theory that precisely characterizes the jumps in the ramification filtration of abelian extensions of local fields, showing they occur at integer values.
-
E.
p-adic numbers
The p-adic numbers are a system of number fields that extend the rational numbers by measuring distance with respect to divisibility by a fixed prime p, playing a central role in modern number theory and arithmetic geometry.
- 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: p-adic Hodge theory Target entity description: 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.
-
A.
Galois representations
Galois representations are homomorphisms from Galois groups of field extensions into matrix groups that encode deep arithmetic information and link number theory with algebraic geometry and modular forms.
-
B.
Iwasawa theory
Iwasawa theory is a branch of number theory that studies the growth of arithmetic invariants in infinite towers of number fields, particularly using p-adic methods.
-
C.
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.
-
D.
Hasse–Arf theorem
The Hasse–Arf theorem is a fundamental result in algebraic number theory that precisely characterizes the jumps in the ramification filtration of abelian extensions of local fields, showing they occur at integer values.
-
E.
p-adic numbers
The p-adic numbers are a system of number fields that extend the rational numbers by measuring distance with respect to divisibility by a fixed prime p, playing a central role in modern number theory and arithmetic geometry.
- F. None of above. chosen
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
ⓘ
linked to:
p-adic Hodge theory
comparison isomorphisms ⓘ p-adic period isomorphisms ⓘ |
| developedBy | Jean-Marc Fontaine ⓘ |
| 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 ⓘ étale cohomology theory ⓘ |
| involvesConstruction |
B_HT
ⓘ
B_cris ⓘ B_dR ⓘ B_st ⓘ Fontaine period rings ⓘ |
| relatedTo |
Iwasawa theory
ⓘ
algebraic geometry ⓘ classical Hodge theory ⓘ motivic cohomology ⓘ number theory ⓘ p-adic Langlands program ⓘ |
| studiesProperty |
Hodge–Tate decomposition
ⓘ
linked to:
p-adic Hodge theory
comparison between different cohomology theories ⓘ crystalline representations ⓘ de Rham representations ⓘ
linked to:
Galois representations
semistable representations ⓘ structure of p-adic Galois representations ⓘ |
| usesConcept |
Galois representations
ⓘ
crystalline cohomology ⓘ de Rham cohomology ⓘ p-adic differential equations ⓘ p-adic fields ⓘ period rings ⓘ rigid cohomology ⓘ étale cohomology ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: p-adic Hodge theory Description of subject: 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.
Referenced by (15)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: p-adic Hodge theory
linked to: p-adic Hodge theory
linked to: p-adic Hodge theory
linked to: p-adic Hodge theory
linked to: p-adic Hodge theory
linked to: p-adic Hodge theory