Shimura varieties
E839486
Shimura varieties are higher-dimensional algebraic varieties that generalize modular curves and play a central role in the Langlands program by connecting number theory, automorphic forms, and arithmetic geometry.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Shimura curve | 1 |
| Shimura varieties canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10062001 — 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: Shimura varieties Context triple: [Hilbert’s twelfth problem, relatedTo, Shimura varieties]
-
A.
Hodge–Riemann bilinear relations
The Hodge–Riemann bilinear relations are fundamental positivity and orthogonality conditions on the intersection form in Hodge theory that underpin results such as the hard Lefschetz theorem and the Hodge index theorem.
-
B.
Standard Conjectures on Algebraic Cycles
The Standard Conjectures on Algebraic Cycles are a set of deep, still unproven hypotheses in algebraic geometry that aim to provide a foundational theory of algebraic cycles and their cohomological properties, underpinning much of the modern theory of motives.
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C.
Hasse–Weil bound for abelian varieties
The Hasse–Weil bound for abelian varieties is a fundamental result in arithmetic geometry that gives sharp estimates for the number of rational points on abelian varieties over finite fields in terms of their dimension and the field size.
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D.
Kummer surfaces
Kummer surfaces are special quartic algebraic surfaces in projective three-space characterized by having 16 ordinary double points, extensively studied in the context of complex geometry and abelian varieties.
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E.
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.
- 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: Shimura varieties Target entity description: Shimura varieties are higher-dimensional algebraic varieties that generalize modular curves and play a central role in the Langlands program by connecting number theory, automorphic forms, and arithmetic geometry.
-
A.
Hodge–Riemann bilinear relations
The Hodge–Riemann bilinear relations are fundamental positivity and orthogonality conditions on the intersection form in Hodge theory that underpin results such as the hard Lefschetz theorem and the Hodge index theorem.
-
B.
Standard Conjectures on Algebraic Cycles
The Standard Conjectures on Algebraic Cycles are a set of deep, still unproven hypotheses in algebraic geometry that aim to provide a foundational theory of algebraic cycles and their cohomological properties, underpinning much of the modern theory of motives.
-
C.
Hasse–Weil bound for abelian varieties
The Hasse–Weil bound for abelian varieties is a fundamental result in arithmetic geometry that gives sharp estimates for the number of rational points on abelian varieties over finite fields in terms of their dimension and the field size.
-
D.
Kummer surfaces
Kummer surfaces are special quartic algebraic surfaces in projective three-space characterized by having 16 ordinary double points, extensively studied in the context of complex geometry and abelian varieties.
-
E.
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.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic variety
ⓘ
higher-dimensional algebraic variety ⓘ object of arithmetic geometry ⓘ |
| appearsIn |
André–Oort conjecture
NERFINISHED
ⓘ
Langlands correspondence NERFINISHED ⓘ Mumford–Tate conjecture NERFINISHED ⓘ Zilber–Pink conjecture NERFINISHED ⓘ |
| definedFrom |
Shimura datum
NERFINISHED
ⓘ
conjugacy class of homomorphisms from Deligne torus ⓘ reductive algebraic group over Q ⓘ |
| generalizes | modular curves ⓘ |
| hasApplication |
study of Hodge structures
ⓘ
study of periods and motives ⓘ study of rational points ⓘ |
| hasComponent |
Shimura variety of Hodge type
NERFINISHED
ⓘ
Shimura variety of PEL type ⓘ Shimura variety of abelian type NERFINISHED ⓘ connected Shimura variety ⓘ |
| hasProperty |
admit minimal (Baily–Borel) compactifications
ⓘ
admit toroidal compactifications ⓘ admits canonical models over reflex field ⓘ can have compactifications ⓘ defined over a number field called reflex field ⓘ often quasi-projective ⓘ |
| hasStructure |
Hecke correspondences
NERFINISHED
ⓘ
Shimura datum NERFINISHED ⓘ canonical model over a number field ⓘ complex analytic uniformization ⓘ special points ⓘ special subvarieties ⓘ |
| namedAfter | Goro Shimura NERFINISHED ⓘ |
| relatedTo |
Galois representations
ⓘ
L-functions NERFINISHED ⓘ Langlands program NERFINISHED ⓘ arithmetic geometry ⓘ automorphic forms ⓘ number theory ⓘ representation theory ⓘ |
| specialCaseOf |
Hilbert modular varieties
NERFINISHED
ⓘ
Siegel modular varieties NERFINISHED ⓘ moduli space of abelian varieties with additional structure ⓘ unitary Shimura varieties NERFINISHED ⓘ |
| studiedBy |
Goro Shimura
NERFINISHED
ⓘ
Pierre Deligne NERFINISHED ⓘ Robert Langlands NERFINISHED ⓘ |
| usedIn |
construction of Galois representations from automorphic forms
ⓘ
formulation of reciprocity laws ⓘ proofs of modularity lifting theorems ⓘ study of motives ⓘ |
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: Shimura varieties Description of subject: Shimura varieties are higher-dimensional algebraic varieties that generalize modular curves and play a central role in the Langlands program by connecting number theory, automorphic forms, and arithmetic geometry.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Shimura curve