Cremona's tables
Generated by GPT-5-miniExpansion Funnel Raw 80 → Dedup 0 → NER 0 → Enqueued 0
| Cremona's tables | |
|---|---|
| Name | Cremona's tables |
| Author | John E. Cremona |
| Country | United Kingdom |
| Language | English |
| Discipline | Mathematics |
| Publisher | University of Warwick |
| Pub date | 1990s–2010s |
Cremona's tables
Cremona's tables are a corpus of computed databases cataloguing isogeny classes of elliptic curves over the rational numbers, compiled and maintained by John E. Cremona and collaborators. Originating from empirical investigations at University of Warwick and later hosted on platforms associated with National Institute of Standards and Technology and the L-functions and Modular Forms Database, the tables connect computational projects in algebraic number theory, modular forms, Galois representations, and explicit instances relevant to conjectures of Birch and Swinnerton-Dyer, Taniyama–Shimura–Weil conjecture, and work by Andrew Wiles. The dataset has informed research at institutions such as University of Cambridge, Princeton University, and Institute for Advanced Study.
Overview and historical context
The project began in the 1980s under the supervision of John E. Cremona at University of Warwick and expanded through collaborations with researchers at University of Bristol, University of Michigan, Harvard University, and University of Oxford. Influenced by prior computational efforts by Noam Elkies, Barry Mazur, and Barry Cipra, the initiative grew alongside algorithmic advances from teams at SageMath, PARI/GP, and MAGMA development groups. Early motivations traced to conjectures by Birch and Swinnerton-Dyer, numerical verifications performed by John Cremona and contemporaries, and the modularity results culminating in the work of Andrew Wiles and Richard Taylor. Funding and dissemination involved agencies like the Science and Engineering Research Council and later digital repositories maintained by the L-functions and Modular Forms Database community.
Mathematical definition and construction
Cremona's tables list elliptic curves over Q organized by conductor, isogeny class, and minimal model, computed using algorithms for integral models developed from ideas of Tate and Silverman. Each entry encodes invariants such as conductor N, Cremona label, rank, regulator, torsion subgroup determined by the theorem of Mazur, real and complex periods related to integrals considered in Weil theory, local Tamagawa numbers following prescriptions by Tate and Milnor, and L-series data connected to work on Hecke operators and modular symbols by Manin and Shokurov. Construction uses descent methods inspired by Selmer group computations and explicit manipulation of Galois cohomology influenced by techniques from Serre and Grothendieck.
Applications in number theory and elliptic curves
The tables serve as empirical evidence for the Birch and Swinnerton-Dyer conjecture, offering datasets for testing predictions on ranks, Tate–Shafarevich groups, and leading coefficients of L-series as explored by Coates and Wiles. They provide concrete examples used in classification problems studied by Mazur and Kubert about torsion subgroups, and in explicit modular parametrizations tied to results of Ribet and Wiles. Researchers in computational arithmetic geometry at Massachusetts Institute of Technology, University of California, Berkeley, and ETH Zurich have used the tables to study regulators, integral points connected to Siegel's theorem, and visibility phenomena examined by Agashe and Stein. The dataset has fed into algorithmic projects by Birch, Swinnerton-Dyer, Mestre, and contemporary investigators at LMS and IMPA.
Computational methods and algorithms
Computations rely on modular symbols algorithms refined by William A. Stein and implementations in SageMath, with heavy use of integer factorization routines from ECM and Lenstra's algorithms, and linear algebra over finite fields leveraged in software from PARI/GP and MAGMA. Rank determination uses two-descent and higher descent methods inspired by Cremona and Tate, combined with analytic rank computations via L-series zeros traced using techniques related to Riemann's explicit formula and computations of Fourier coefficients of newforms per Atkin and Lehner. Data validation adopted modularity criteria derived from Carayol and numeric verification protocols employed by Rubin and Silverman. High-throughput computations exploited clusters at University of Warwick and distributed contributions coordinated with projects at Centre for Number Theory and Mathematical Sciences Research Institute.
Examples and notable tables
Notable entries include small-conductor curves that illustrated unexpected ranks studied by Noam Elkies and examples where Tate–Shafarevich groups were computed or conjectured nontrivial by investigators such as Don Zagier and Brian Birch. Specific conductors like 11, 37, 43 appear as canonical examples tied to classical modular curves from work by Fricke and Klein, while higher-conductor families highlighted in papers by Cremona, Mestre, and Brumer revealed phenomena relevant to Goldfeld's conjecture. The tables are cited in computational records curated by LMFDB and used in expository treatments by Silverman and Cohen.
Influence and subsequent developments
Cremona's work catalyzed community efforts resulting in the L-functions and Modular Forms Database and influenced development of computational packages by William A. Stein, John Cremona, and contributors to SageMath and PARI/GP. The methodology informed later projects on higher-dimensional abelian varieties studied at Princeton University and Harvard University, and inspired algorithmic research by Brumer and McGuinness. The legacy persists in datasets used in doctoral research at University of Cambridge, collaborative programs at Simons Foundation-funded institutes, and in applied contexts interfacing with cryptographic research at NIST and industry labs.