LMFDB

Welcome to the LMFDB, the database of L-functions, modular forms, and related objects. These pages are intended to be a modern handbook including tables, formulas, links, references, etc. to very concrete objects, in particular specific L-functions and their sources. L-functions are ubiquitous in number theory and have applications to mathematical physics and cryptography. By an L-function, we generally mean a Dirichlet series with a functional equation and an Euler product, the simplest example being the Riemann zeta function. Two of the seven Clay Mathematics Million Dollar Millennium Problems deal with properties of these functions, namely the Riemann Hypothesis and the Birch and Swinnerton-Dyer Conjecture. L-functions arise from and encode information about a number of mathematical objects. It is necessary to exhibit these objects along with the L-functions themselves, since typically we need these objects to compute L-functions. In these pages you will see examples of L-functions coming from modular forms, elliptic curves, number fields, and Dirichlet characters, as well as more generally from automorphic forms, algebraic varieties, and Artin representations. In addition, the database contains details about these objects themselves. See the Map of LMFDB for descriptions of connections between these objects. For additional information, there is a useful collection of freely available online sources at http://www.numbertheory.org/ntw/lecture_notes.html. The subject of L-functions is very rich, with many interrelationships. Our goal is to describe the data in ways that faithfully exhibit these interconnections, and to offer access to the data as a means of prompting further exploration and discovery. We believe that the creation of this website will lead to the development and understanding of new mathematics.


References in zbMATH (referenced in 79 articles )

Showing results 1 to 20 of 79.
Sorted by year (citations)

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  1. Daniels, Harris B.; González-Jiménez, Enrique: On the torsion of rational elliptic curves over sextic fields (2020)
  2. Ahlgren, Scott; Dunn, Alexander: Maass forms and the mock theta function (f(q)) (2019)
  3. Bary-Soroker, Lior; Stix, Jakob: Cubic twin prime polynomials are counted by a modular form (2019)
  4. Bennett, Michael A.; Gherga, Adela; Rechnitzer, Andrew: Computing elliptic curves over (\mathbbQ) (2019)
  5. Bergdall, John; Pollack, Robert: Slopes of modular forms and the ghost conjecture. II (2019)
  6. Billerey, Nicolas; Chen, Imin; Dieulefait, Luis; Freitas, Nuno: A multi-frey approach to Fermat equations of signature ((r,r,p)) (2019)
  7. Costa, Edgar; Mascot, Nicolas; Sijsling, Jeroen; Voight, John: Rigorous computation of the endomorphism ring of a Jacobian (2019)
  8. Cremona, John; Pacetti, Ariel: On elliptic curves of prime power conductor over imaginary quadratic fields with class number 1 (2019)
  9. Daniels, Harris B.; Derickx, Maarten; Hatley, Jeffrey: Groups of generalized $G$-type and applications to torsion subgroups of rational elliptic curves over infinite extensions of $\mathbbQ$ (2019)
  10. Derickx, Maarten; Najman, Filip: Torsion of elliptic curves over cyclic cubic fields (2019)
  11. Dokchitser, Tim; Doris, Christopher: 3-torsion and conductor of genus 2 curves (2019)
  12. Earnest, A. G.; Haensch, Anna: Classification of one-class spinor genera for quaternary quadratic forms (2019)
  13. Goldfeld, Dorian; Huang, Bingrong: Super-positivity of a family of L-functions (2019)
  14. Gu, Miao; Martin, Greg: Factorization tests and algorithms arising from counting modular forms and automorphic representations (2019)
  15. Jenkins, Robert; McLaughlin, Ken D. T.-R.: Behavior of the roots of the Taylor polynomials of Riemann’s (\xi) function with growing degree (2019)
  16. Kazalicki, Matija: Congruences for sporadic sequences and modular forms for non-congruence subgroups (2019)
  17. Lairez, Pierre; Sertöz, Emre Can: A numerical transcendental method in algebraic geometry: computation of Picard groups and related invariants (2019)
  18. Lemos, Pedro: Serre’s uniformity conjecture for elliptic curves with rational cyclic isogenies (2019)
  19. Lemos, Pedro: Some cases of Serre’s uniformity problem (2019)
  20. Li, Wen-Ching Winnie; Liu, Tong; Long, Ling: Potentially (\operatornameGL_2)-type Galois representations associated to noncongruence modular forms (2019)

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