Deuring reduction theorem
E204743
The Deuring reduction theorem is a result in number theory that relates the reduction of elliptic curves with complex multiplication modulo primes to the theory of quaternion algebras and endomorphism rings.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Deuring reduction theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1822598 — 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: Deuring reduction theorem Context triple: [Max Deuring, notableWork, Deuring reduction theorem]
-
A.
Hilbert’s syzygy theorem
Hilbert’s syzygy theorem is a fundamental result in commutative algebra that describes the finite length and structure of free resolutions of modules over polynomial rings.
-
B.
Hilbert’s irreducibility theorem
Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
-
C.
Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
-
D.
Riemann–Roch theorem
The Riemann–Roch theorem is a fundamental result in algebraic geometry and complex analysis that relates the dimension of spaces of meromorphic sections of a line bundle on a curve to topological data such as genus and degree.
-
E.
Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
- 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: Deuring reduction theorem Target entity description: The Deuring reduction theorem is a result in number theory that relates the reduction of elliptic curves with complex multiplication modulo primes to the theory of quaternion algebras and endomorphism rings.
-
A.
Hilbert’s syzygy theorem
Hilbert’s syzygy theorem is a fundamental result in commutative algebra that describes the finite length and structure of free resolutions of modules over polynomial rings.
-
B.
Hilbert’s irreducibility theorem
Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
-
C.
Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
-
D.
Riemann–Roch theorem
The Riemann–Roch theorem is a fundamental result in algebraic geometry and complex analysis that relates the dimension of spaces of meromorphic sections of a line bundle on a curve to topological data such as genus and degree.
-
E.
Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
- F. None of above. chosen
Statements (41)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
theorem in number theory ⓘ |
| appliesTo | elliptic curves with complex multiplication ⓘ |
| characterizes |
when reduction of a CM elliptic curve is ordinary
ⓘ
when reduction of a CM elliptic curve is supersingular ⓘ |
| concerns |
primes of good reduction
ⓘ
reduction of j-invariants modulo primes ⓘ |
| describes |
correspondence between CM elliptic curves and ideals in orders of imaginary quadratic fields
ⓘ
correspondence between supersingular elliptic curves and maximal orders in quaternion algebras ⓘ how endomorphism rings change under reduction modulo primes ⓘ |
| field |
arithmetic geometry
ⓘ
number theory ⓘ |
| hasConsequence |
classification of supersingular j-invariants in characteristic p
ⓘ
description of endomorphism rings of supersingular elliptic curves ⓘ link between CM theory and supersingular theory ⓘ |
| historicalPeriod | 20th century mathematics ⓘ |
| involves |
definite quaternion algebras over the rationals
ⓘ
imaginary quadratic fields ⓘ ordinary elliptic curves ⓘ supersingular elliptic curves ⓘ |
| isUsedIn |
explicit construction of class fields via CM elliptic curves
ⓘ
local and global class field theory for CM fields ⓘ study of Galois representations attached to elliptic curves ⓘ study of modular curves ⓘ theory of abelian varieties with complex multiplication ⓘ theory of supersingular isogeny graphs ⓘ |
| languageOfOriginalPublication | German ⓘ |
| mainSubject |
complex multiplication
ⓘ
elliptic curves ⓘ endomorphism rings ⓘ quaternion algebras ⓘ reduction modulo primes ⓘ |
| namedAfter | Max Deuring ⓘ |
| relates |
CM elliptic curves over number fields
ⓘ
endomorphism rings of elliptic curves in characteristic p ⓘ endomorphism rings of elliptic curves in characteristic zero ⓘ maximal orders in quaternion algebras ⓘ reductions of elliptic curves modulo primes ⓘ |
| uses |
ideal class groups
ⓘ
theory of complex multiplication ⓘ theory of quaternion algebras ⓘ |
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: Deuring reduction theorem Description of subject: The Deuring reduction theorem is a result in number theory that relates the reduction of elliptic curves with complex multiplication modulo primes to the theory of quaternion algebras and endomorphism rings.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.