Triple

T17671650
Position Surface form Disambiguated ID Type / Status
Subject Benedict Gross E440533 entity
Predicate notableWork P4 FINISHED
Object Gross–Zagier formula
The Gross–Zagier formula is a fundamental result in number theory that relates the heights of Heegner points on elliptic curves to the derivatives of associated L-functions, with deep implications for the Birch and Swinnerton-Dyer conjecture.
E1281172 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–Zagier formula | Statement: [Benedict Gross, notableWork, Gross–Zagier formula]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Gross–Zagier formula
Context triple: [Benedict Gross, notableWork, Gross–Zagier formula]
  • A. Bloch–Kato conjecture
    The Bloch–Kato conjecture is a deep statement in arithmetic geometry and K-theory that predicts an exact correspondence between Galois cohomology and Milnor K-theory, linking algebraic K-groups to field arithmetic.
  • B. Siegel’s theorem on zeros of L-functions
    Siegel’s theorem on zeros of L-functions is a result in analytic number theory that gives strong bounds on how close nontrivial zeros of Dirichlet L-functions can approach 1, with deep implications for the distribution of primes in arithmetic progressions.
  • C. 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.
  • D. Ribet's theorem
    Ribet's theorem is a result in number theory that linked certain modular forms to Galois representations and played a crucial role in the proof of Fermat's Last Theorem.
  • E. Fontaine–Mazur conjecture
    The Fontaine–Mazur conjecture is a central open problem in number theory that predicts which p-adic Galois representations of number fields arise from geometry or from automorphic forms.
  • 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–Zagier formula
Triple: [Benedict Gross, notableWork, Gross–Zagier formula]
Generated description
The Gross–Zagier formula is a fundamental result in number theory that relates the heights of Heegner points on elliptic curves to the derivatives of associated L-functions, with deep implications for the Birch and Swinnerton-Dyer conjecture.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Gross–Zagier formula
Target entity description: The Gross–Zagier formula is a fundamental result in number theory that relates the heights of Heegner points on elliptic curves to the derivatives of associated L-functions, with deep implications for the Birch and Swinnerton-Dyer conjecture.
  • A. Bloch–Kato conjecture
    The Bloch–Kato conjecture is a deep statement in arithmetic geometry and K-theory that predicts an exact correspondence between Galois cohomology and Milnor K-theory, linking algebraic K-groups to field arithmetic.
  • B. Siegel’s theorem on zeros of L-functions
    Siegel’s theorem on zeros of L-functions is a result in analytic number theory that gives strong bounds on how close nontrivial zeros of Dirichlet L-functions can approach 1, with deep implications for the distribution of primes in arithmetic progressions.
  • C. 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.
  • D. Ribet's theorem
    Ribet's theorem is a result in number theory that linked certain modular forms to Galois representations and played a crucial role in the proof of Fermat's Last Theorem.
  • E. Fontaine–Mazur conjecture
    The Fontaine–Mazur conjecture is a central open problem in number theory that predicts which p-adic Galois representations of number fields arise from geometry or from automorphic forms.
  • 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.