Triple

T17671651
Position Surface form Disambiguated ID Type / Status
Subject Benedict Gross E440533 entity
Predicate notableWork P4 FINISHED
Object Gross–Koblitz formula
The Gross–Koblitz formula is a result in number theory that expresses Gauss sums in terms of the p-adic gamma function, linking exponential sums over finite fields with p-adic analysis.
E1281173 NE FINISHED

How this triple was built (4 steps)

Every LLM step that produced this triple, in pipeline order — named-entity classification, the disambiguation choices (the exact options shown, with the pick highlighted), and the generated description. The batch + timestamp of each is in the Provenance table below.

NER Named-entity recognition gpt-5-mini
Instruction
Given a phrase, classify it is english named entity (e.g., persons, organizations, works of art) in Latin script, or not (e.g., literals, dates, URLs, verbose phrases). For disambiguation, the statement where the phrase occurs as object is also given. Please return a JSON object with `phrase` (string, the phrase being analyzed) and `is_ne` (boolean, indicating whether the phrase is a Named Entity).
Input
Phrase: Gross–Koblitz formula | Statement: [Benedict Gross, notableWork, Gross–Koblitz formula]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Gross–Koblitz formula
Context triple: [Benedict Gross, notableWork, Gross–Koblitz formula]
  • A. 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.
  • B. Kummer congruences
    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.
  • C. Lubin–Tate formal groups
    Lubin–Tate formal groups are a class of one-dimensional formal group laws over local fields that play a central role in local class field theory by providing explicit descriptions of abelian extensions.
  • D. Drinfeld modules
    Drinfeld modules are algebraic structures that generalize elliptic curves to the setting of function fields, playing a central role in modern arithmetic geometry and the theory of automorphic forms.
  • E. Grothendieck–Lefschetz trace formula
    The Grothendieck–Lefschetz trace formula is a fundamental result in algebraic geometry that expresses the number of rational points of a variety over a finite field in terms of traces of Frobenius acting on its étale cohomology groups.
  • F. None of above. chosen
  • G. Unsure - the case is ambiguous/there is not enough information to decide.
NEDg Description generation gpt-5.1
Instruction
Generate a one-sentence description of the target entity. 
You are given a context triple in the form (subject, predicate, object), where the object is the target entity. 
# Instructions
Use the triple to infer relevant information about the entity. Describe the entity based on what is most defining, well-known. 
Avoid repeating the information from the triple, unless really essential.
# Response Format
Return only the sentence: "Description: [one-sentence description of the target entity]"
Input
Entity: Gross–Koblitz formula
Triple: [Benedict Gross, notableWork, Gross–Koblitz formula]
Generated description
The Gross–Koblitz formula is a result in number theory that expresses Gauss sums in terms of the p-adic gamma function, linking exponential sums over finite fields with p-adic analysis.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Gross–Koblitz formula
Target entity description: The Gross–Koblitz formula is a result in number theory that expresses Gauss sums in terms of the p-adic gamma function, linking exponential sums over finite fields with p-adic analysis.
  • A. 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.
  • B. Kummer congruences
    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.
  • C. Lubin–Tate formal groups
    Lubin–Tate formal groups are a class of one-dimensional formal group laws over local fields that play a central role in local class field theory by providing explicit descriptions of abelian extensions.
  • D. Drinfeld modules
    Drinfeld modules are algebraic structures that generalize elliptic curves to the setting of function fields, playing a central role in modern arithmetic geometry and the theory of automorphic forms.
  • E. Grothendieck–Lefschetz trace formula
    The Grothendieck–Lefschetz trace formula is a fundamental result in algebraic geometry that expresses the number of rational points of a variety over a finite field in terms of traces of Frobenius acting on its étale cohomology groups.
  • F. None of above. chosen

Provenance (5 batches)

The batch behind each pipeline step, in order, with when it ran. Timestamps are batch-level — stages were processed in waves, so the object chain (NER → NED1 → NEDg → NED2) reads in order, but predicate / elicitation batches can sit in a different wave.

Step Stage Batch ID Status When
creating Elicitation batch_69d8b9e87e18819087104a44dc4dc5b1 completed April 10, 2026, 8:50 a.m.
NER Named-entity recognition batch_69e46f69b11c8190b09add33f81776b3 completed April 19, 2026, 6 a.m.
NED1 Entity disambiguation (via context triple) batch_6a02166234f0819099c452808234e3e4 completed May 11, 2026, 5:48 p.m.
NEDg Description generation batch_6a0217ca2ec881908573b0423c3610f7 completed May 11, 2026, 5:54 p.m.
NED2 Entity disambiguation (via description) batch_6a0218636e048190a48bc7f066c7bee1 completed May 11, 2026, 5:56 p.m.
Created at: April 10, 2026, 9:59 a.m.