Kummer congruences
E463786
Kummer congruences are number-theoretic relations describing how special values of Bernoulli numbers and related arithmetic functions behave modulo powers of primes, foundational in the study of p-adic L-functions and cyclotomic fields.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Kummer congruences canonical | 2 |
| Kummer’s criterion for Fermat’s Last Theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T4706379 — 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: Kummer congruences Context triple: [Ernst Eduard Kummer, knownFor, Kummer congruences]
-
A.
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.
-
B.
Ono’s partition congruences
Ono’s partition congruences are modern number-theoretic results that extend Ramanujan’s classical congruences by proving the existence of infinitely many congruence relations for the partition function modulo various primes.
-
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.
Ramanujan–Petersson conjecture
The Ramanujan–Petersson conjecture is a fundamental statement in number theory and the theory of modular forms that predicts strong bounds on the Fourier coefficients of modular cusp forms, with deep connections to automorphic forms and the Langlands program.
-
E.
Ramanujan partition congruences
Ramanujan partition congruences are remarkable number-theoretic results discovered by Srinivasa Ramanujan that describe surprising modular patterns in the partition function, such as specific arithmetic progressions where the number of integer partitions of an integer is divisible by a given 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: Kummer congruences Target entity description: Kummer congruences are number-theoretic relations describing how special values of Bernoulli numbers and related arithmetic functions behave modulo powers of primes, foundational in the study of p-adic L-functions and cyclotomic fields.
-
A.
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.
-
B.
Ono’s partition congruences
Ono’s partition congruences are modern number-theoretic results that extend Ramanujan’s classical congruences by proving the existence of infinitely many congruence relations for the partition function modulo various primes.
-
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.
Ramanujan–Petersson conjecture
The Ramanujan–Petersson conjecture is a fundamental statement in number theory and the theory of modular forms that predicts strong bounds on the Fourier coefficients of modular cusp forms, with deep connections to automorphic forms and the Langlands program.
-
E.
Ramanujan partition congruences
Ramanujan partition congruences are remarkable number-theoretic results discovered by Srinivasa Ramanujan that describe surprising modular patterns in the partition function, such as specific arithmetic progressions where the number of integer partitions of an integer is divisible by a given prime.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical concept
ⓘ
number-theoretic congruence ⓘ result in algebraic number theory ⓘ |
| appearsIn |
proofs of properties of Bernoulli numbers modulo primes
ⓘ
theory of regular and irregular primes ⓘ |
| appliesTo |
Bernoulli numbers B_n
ⓘ
linked to:
Bernoulli numbers
special values of Dirichlet L-functions at non-positive integers ⓘ special values of the Riemann zeta function at negative integers ⓘ |
| characterizes |
p-adic interpolation of Bernoulli numbers
ⓘ
p-adic interpolation of special L-values ⓘ |
| describes |
behavior of Bernoulli numbers modulo p^n
ⓘ
congruence relations between Bernoulli numbers modulo powers of primes ⓘ p-adic continuity of special values of L-functions ⓘ |
| field | number theory ⓘ |
| formalizes | compatibility of special L-values in p-adic families ⓘ |
| generalizedBy |
Iwasawa theory congruences
ⓘ
p-adic interpolation theorems ⓘ |
| hasConsequence |
criteria for irregular primes
ⓘ
relations among class numbers of cyclotomic fields at different levels ⓘ |
| historicalPeriod | 19th century mathematics ⓘ |
| holdsFor | even indices of Bernoulli numbers ⓘ |
| introducedBy | Ernst Eduard Kummer ⓘ |
| involves |
congruences modulo p^k
ⓘ
p-adic valuations ⓘ powers of primes p^n ⓘ prime numbers p ⓘ |
| motivation |
study of Fermat's Last Theorem for regular primes
ⓘ
study of class numbers of cyclotomic fields ⓘ |
| namedAfter | Ernst Eduard Kummer ⓘ |
| relatedTo |
Bernoulli numbers
ⓘ
Dirichlet L-functions ⓘ Fermat's Last Theorem ⓘ Iwasawa main conjecture ⓘ
linked to:
Iwasawa theory
Kubota–Leopoldt p-adic L-functions ⓘ
linked to:
p-adic L-functions
class number formula ⓘ cyclotomic fields ⓘ ideal class groups ⓘ irregular primes ⓘ p-adic L-functions ⓘ special values of L-functions ⓘ |
| subfield |
Iwasawa theory
ⓘ
algebraic number theory ⓘ p-adic number theory ⓘ |
| usedIn |
Iwasawa theory of cyclotomic fields
ⓘ
construction of p-adic L-functions ⓘ proofs of properties of p-adic zeta functions ⓘ study of class groups of cyclotomic fields ⓘ study of cyclotomic extensions of Q ⓘ |
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: Kummer congruences Description of subject: Kummer congruences are number-theoretic relations describing how special values of Bernoulli numbers and related arithmetic functions behave modulo powers of primes, foundational in the study of p-adic L-functions and cyclotomic fields.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Kummer congruences