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 54 articles )

Showing results 1 to 20 of 54.
Sorted by year (citations)

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  1. Bennett, Michael A.; Gherga, Adela; Rechnitzer, Andrew: Computing elliptic curves over (\mathbbQ) (2019)
  2. Costa, Edgar; Mascot, Nicolas; Sijsling, Jeroen; Voight, John: Rigorous computation of the endomorphism ring of a Jacobian (2019)
  3. Dokchitser, Tim; Doris, Christopher: 3-torsion and conductor of genus 2 curves (2019)
  4. Gu, Miao; Martin, Greg: Factorization tests and algorithms arising from counting modular forms and automorphic representations (2019)
  5. Jenkins, Robert; McLaughlin, Ken D. T.-R.: Behavior of the roots of the Taylor polynomials of Riemann’s (\xi) function with growing degree (2019)
  6. Lemos, Pedro: Serre’s uniformity conjecture for elliptic curves with rational cyclic isogenies (2019)
  7. Lombardo, Davide: Computing the geometric endomorphism ring of a genus-2 Jacobian (2019)
  8. Molin, Pascal; Neurohr, Christian: Computing period matrices and the Abel-Jacobi map of superelliptic curves (2019)
  9. Wood, Melanie Matchett: Nonabelian Cohen-Lenstra moments (2019)
  10. Bruin, Peter; Ferraguti, Andrea: On (L)-functions of quadratic (\mathbbQ)-curves (2018)
  11. Creutz, Brendan: Improved rank bounds from (2)-descent on hyperelliptic Jacobians (2018)
  12. Daniels, Harris B.; Lozano-Robledo, Álvaro; Najman, Filip; Sutherland, Andrew V.: Torsion subgroups of rational elliptic curves over the compositum of all cubic fields (2018)
  13. Elder, G. Griffith: Ramified extensions of degree (p) and their Hopf-Galois module structure (2018)
  14. Fité, Francesc; Guitart, Xavier: Fields of definition of elliptic (k)-curves and the realizability of all genus 2 Sato-Tate groups over a number field (2018)
  15. Hildebrand, A. J.: Unexpected regularities in the behavior of some number-theoretic power series (2018)
  16. Jones, John W.; Roberts, David P.: Mixed degree number field computations (2018)
  17. Kamienny, Sheldon; Newman, Burton: Points of order 13 on elliptic curves (2018)
  18. Kiming, Ian; Rustom, Nadim: Dihedral group, 4-torsion on an elliptic curve, and a peculiar eigenform modulo 4 (2018)
  19. Kohen, Daniel; Pacetti, Ariel: On Heegner points for primes of additive reduction ramifying in the base field (2018)
  20. Lapkova, Kostadinka: Explicit upper bound for the average number of divisors of irreducible quadratic polynomials (2018)

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