Stark conjectures
E839485
The Stark conjectures are a set of deep conjectures in algebraic number theory that predict precise connections between special values of L-functions and the arithmetic of number fields, particularly units and class fields.
All labels observed (7)
| Label | Occurrences |
|---|---|
| Brumer–Stark conjecture | 2 |
| Stark conjectures canonical | 2 |
| Gross–Stark conjecture | 1 |
| Rubin–Stark conjecture | 1 |
| Stark conjecture for totally real fields | 1 |
| higher-rank Stark conjectures | 1 |
| rank-one Stark conjecture | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10061997 — 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: Stark conjectures Context triple: [Hilbert’s twelfth problem, relatedTo, Stark conjectures]
-
A.
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.
-
B.
Beilinson conjectures
Beilinson conjectures are a set of deep conjectures in arithmetic geometry that relate special values of L-functions to algebraic K-theory and motivic cohomology, generalizing phenomena seen in cases like the Birch and Swinnerton-Dyer conjecture.
-
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.
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.
-
E.
Artin’s conjecture on L-functions
Artin’s conjecture on L-functions is a major unproven hypothesis in number theory asserting that nontrivial Artin L-functions associated to Galois representations are entire, with deep implications for the distribution of primes and the structure of number fields.
- 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: Stark conjectures Target entity description: The Stark conjectures are a set of deep conjectures in algebraic number theory that predict precise connections between special values of L-functions and the arithmetic of number fields, particularly units and class fields.
-
A.
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.
-
B.
Beilinson conjectures
Beilinson conjectures are a set of deep conjectures in arithmetic geometry that relate special values of L-functions to algebraic K-theory and motivic cohomology, generalizing phenomena seen in cases like the Birch and Swinnerton-Dyer conjecture.
-
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.
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.
-
E.
Artin’s conjecture on L-functions
Artin’s conjecture on L-functions is a major unproven hypothesis in number theory asserting that nontrivial Artin L-functions associated to Galois representations are entire, with deep implications for the distribution of primes and the structure of number fields.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
conjecture in algebraic number theory
ⓘ
mathematical conjecture ⓘ |
| concerns |
leading terms of Artin L-functions at s = 0
ⓘ
special values of L-functions at s = 0 ⓘ special values of L-functions at s = 1 ⓘ |
| field | algebraic number theory ⓘ |
| goal |
describe arithmetic invariants via L-values
ⓘ
give explicit generators of abelian extensions ⓘ |
| hasPart |
Stark conjecture for totally real fields
ⓘ
linked to:
Stark conjectures
Stark conjecture over Q ⓘ higher-rank Stark conjectures ⓘ
linked to:
Stark conjectures
rank-one Stark conjecture ⓘ
linked to:
Stark conjectures
|
| implies |
existence of canonical units in certain extensions
ⓘ
explicit class field theory statements ⓘ |
| influenceOn |
development of modern class field theory
ⓘ
research on special values of L-functions ⓘ |
| involves |
Artin characters
ⓘ
Dirichlet characters ⓘ Galois representations ⓘ idele class groups ⓘ regulators of units ⓘ |
| mainTheme |
arithmetic of number fields
ⓘ
class fields ⓘ special values of L-functions ⓘ units in number fields ⓘ |
| namedAfter | Harold Stark ⓘ |
| predicts |
connections between special L-values and units
ⓘ
construction of abelian extensions from L-values ⓘ existence of Stark units ⓘ relations between regulators and derivatives of L-functions ⓘ |
| proposedBy | Harold Stark ⓘ |
| relatedTo |
Birch and Swinnerton-Dyer conjecture
ⓘ
linked to:
Birch and Swinnerton-Dyer Conjecture
Brumer–Stark conjecture ⓘ
linked to:
Stark conjectures
Gross–Stark conjecture ⓘ
linked to:
Stark conjectures
Leopoldt conjecture ⓘ Rubin–Stark conjecture ⓘ
linked to:
Stark conjectures
|
| relates |
Artin L-functions
ⓘ
L-functions ⓘ abelian extensions of number fields ⓘ class field theory ⓘ |
| status | open problem in mathematics ⓘ |
| timePeriod | 20th century ⓘ |
| usedIn |
Iwasawa theory
ⓘ
construction of p-adic L-functions ⓘ explicit class field theory ⓘ |
How these facts were elicited
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Input
Subject: Stark conjectures Description of subject: The Stark conjectures are a set of deep conjectures in algebraic number theory that predict precise connections between special values of L-functions and the arithmetic of number fields, particularly units and class fields.
Referenced by (9)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Stark conjectures
linked to: Stark conjectures
linked to: Stark conjectures
linked to: Stark conjectures
linked to: Stark conjectures
linked to: Stark conjectures
linked to: Stark conjectures