Iwasawa theory
E358023
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Iwasawa theory canonical | 16 |
| Iwasawa main conjecture | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T3424532 — 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: Iwasawa theory Context triple: [Karl Rubin, researchArea, Iwasawa theory]
-
A.
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.
-
B.
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.
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C.
Hilbert’s twelfth problem
Hilbert’s twelfth problem is one of David Hilbert’s famous list of 23 problems, asking for a general explicit class field theory that would generate all abelian extensions of a given number field using special values of analytic functions.
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D.
Dedekind zeta functions
Dedekind zeta functions are number-theoretic functions attached to algebraic number fields that encode their arithmetic properties, such as the distribution of prime ideals and class numbers.
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E.
Algebraic Groups and Class Fields
"Algebraic Groups and Class Fields" is a influential mathematical monograph that develops the deep connections between algebraic group theory and class field theory within 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: Iwasawa theory Target entity description: 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.
-
A.
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.
-
B.
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.
-
C.
Hilbert’s twelfth problem
Hilbert’s twelfth problem is one of David Hilbert’s famous list of 23 problems, asking for a general explicit class field theory that would generate all abelian extensions of a given number field using special values of analytic functions.
-
D.
Dedekind zeta functions
Dedekind zeta functions are number-theoretic functions attached to algebraic number fields that encode their arithmetic properties, such as the distribution of prime ideals and class numbers.
-
E.
Algebraic Groups and Class Fields
"Algebraic Groups and Class Fields" is a influential mathematical monograph that develops the deep connections between algebraic group theory and class field theory within number theory and arithmetic geometry.
- F. None of above. chosen
Statements (51)
| Predicate | Object |
|---|---|
| instanceOf |
branch of number theory
ⓘ
mathematical theory ⓘ |
| appliesTo |
class groups of number fields
ⓘ
elliptic curves over number fields ⓘ ideal class groups of cyclotomic fields ⓘ modular forms ⓘ motives ⓘ |
| coreConcept |
Greenberg Selmer group
ⓘ
linked to:
Selmer group
Iwasawa algebra ⓘ Iwasawa invariants ⓘ Iwasawa module ⓘ Mazur control theorem ⓘ Selmer group ⓘ Z_p-extension ⓘ control theorem ⓘ cyclotomic Z_p-extension ⓘ lambda invariant ⓘ main conjecture of Iwasawa theory ⓘ mu invariant ⓘ nu invariant ⓘ |
| developedBy |
Kenkichi Iwasawa
ⓘ
linked to:
Kenku Iwasawa
|
| field | number theory ⓘ |
| focusesOn |
Galois groups of infinite pro-p extensions
ⓘ
Z_p-extensions of number fields ⓘ structure of modules over Iwasawa algebras ⓘ |
| furtherDevelopedBy |
Andrew Wiles
ⓘ
Barry Mazur ⓘ Jean-Pierre Serre ⓘ John Coates ⓘ Ralph Greenberg ⓘ |
| namedAfter |
Kenkichi Iwasawa
ⓘ
linked to:
Kenku Iwasawa
Kenku Iwasawa ⓘ |
| relatedTo |
Birch and Swinnerton-Dyer conjecture
ⓘ
linked to:
Birch and Swinnerton-Dyer Conjecture
Bloch–Kato conjecture ⓘ Galois cohomology ⓘ Langlands program ⓘ algebraic number theory ⓘ arithmetic geometry ⓘ p-adic Hodge theory ⓘ |
| studies |
Galois modules
ⓘ
Galois representations ⓘ Selmer groups in towers of number fields ⓘ arithmetic of elliptic curves in towers of fields ⓘ arithmetic of modular forms in towers of fields ⓘ class groups in towers of number fields ⓘ cyclotomic extensions ⓘ growth of arithmetic invariants ⓘ infinite towers of number fields ⓘ p-adic L-functions ⓘ p-primary parts of class groups ⓘ |
| usesMethod | p-adic methods ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Iwasawa theory Description of subject: 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.
Referenced by (17)
Full triples — surface form annotated when it differs from this entity's canonical label.