Source-linked AI summary
The L-functions and modular forms database project
John Cremona
TL;DR
The project addresses the difficulty of understanding relationships among specialized mathematical objects and their L-functions. It develops and organizes the LMFDB as a systematic, concrete database and website, with linked data and open-source collaboration. The resulting resource displays interrelations among objects and supports computation, classification, education, and research, while some defining properties remain unproved and technical support is limited.
Problem
Mathematical objects connected to L-functions are highly specialized, while their relationships and available data remain scattered and incomplete.
Method
The project organizes L-functions and modular forms in the LMFDB through linked databases, a website, open-source software, and collaborative development.
Results
The LMFDB displays interrelations among linked mathematical objects and provides data, code, visualizations, search, and knowledge resources for further work.
Takeaways & Limitations
The LMFDB serves as a pedagogical and research resource for exploring, computing with, and studying L-functions and related objects.
Takeaways & Limitations
Some defining L-function properties remain unproved, and the project lacks dedicated technical support, relying partly on charitable contributions of time.
Abstract
from arXiv · showhide
The Langlands Programme, formulated by Robert Langlands in the 1960s and since much developed and refined, is a web of interrelated theory and conjectures concerning many objects in number theory, their interconnections, and connections to other fields. At the heart of the Langlands Programme is the concept of an L-function. The most famous L-function is the Riemann zeta-function, and as well as being ubiquitous in number theory itself, L-functions have applications in mathematical physics and cryptography. Two of the seven Clay Mathematics Million Dollar Millennium Problems, the Riemann Hypothesis and the Birch and Swinnerton-Dyer Conjecture, deal with their properties. Many different mathematical objects are connected in various ways to L-functions, but the study of those objects is highly specialized, and most mathematicians have only a vague idea of the objects outside their specialty and how everything is related. Helping mathematicians to understand these connections was the motivation for the L-functions and Modular Forms Database (LMFDB) project. Its mission is to chart the landscape of L-functions and modular forms in a systematic, comprehensive and concrete fashion. This involves developing their theory, creating and improving algorithms for computing and classifying them, and hence discovering new properties of these functions, and testing fundamental conjectures. In the lecture I gave a very brief introduction to L-functions for non-experts, and explained and demonstrated how the large collection of data in the LMFDB is organized and displayed, showing the interrelations between linked objects, through our website www.lmfdb.org. I also showed how this has been created by a world-wide open source collaboration, which we hope may become a model for others.
1. What is the LMFDB?
The LMFDB addresses scattered, incomplete, and inconsistent mathematical databases by bringing data together in a substantially improved resource.
- Before the LMFDB, number-theory tables and databases were difficult to use, scattered across personal webpages, and presented in inconsistent formats.Users often had to identify the relevant owner, download data, and handle different interfaces and formats.
- Even where software packages provided elliptic-curve databases, the data remained scattered and incomplete.
- The LMFDB substantially improves access to this landscape of mathematical data.
2. L-functions and why they are important
L-functions are complex analytic functions characterized by series, products, and functional equations, with the Riemann zeta function as the simplest example and links to prime distribution.
- L-functions are central to the LMFDB and are introduced through a brief survey for non-experts.
- The Riemann zeta function is presented as the simplest L-function.
- L-functions are connected to the distribution of primes.
- An L-function has a Dirichlet series, an Euler product, and a functional equation, alongside additional technical axioms.The database’s definition omits the further Selberg axioms for this introductory account.
2.1. L-functions: a definition.
L-functions encode analytic properties of mathematical objects, but some defining properties remain unproved, while conjectures about their zeros have major mathematical significance.
- Some defining L-function properties remain unproved for objects in the database, including elliptic curves over general algebraic number fields.Modularity of elliptic curves over Q was proved by Wiles, whereas the corresponding general result over other number fields is not yet known.
- The Riemann hypothesis has remained open since Riemann formulated it in 1859 and is a Clay Mathematics Institute Millennium Prize Problem.
- The Riemann hypothesis asserts that every non-trivial zero of ζ(s) lies on the critical line ℜ(s) = 1/2.
- Generalized Riemann Hypotheses concern zero locations for all L-functions and affect the computational complexity of important number-theoretic quantities.For example, assuming GRH can make computation of a number field’s class number much faster.
- The LMFDB provides pedagogical pages and graphs for the Riemann zeta function, including the first few zeros along the critical line.
- The database stores more than 10^11 explicitly computed zeta zeros at 100-bit precision for studying their distribution and connections to random matrices.
2.3. Degrees of L-functions.
The Euler product encodes an L-function through local polynomials whose common degree defines the L-function’s degree, while exceptional primes are controlled by the conductor. The Riemann zeta-function provides the basic degree-one example.
- Degrees of L-functions: Each local polynomial Pp(t) has the common degree called the degree of the L-function, except at finitely many primes dividing its conductor.At conductor-dividing primes, the polynomial degree is smaller.
- Degrees of L-functions: ζ(s) has Pp(t) = 1 − t for every prime, degree d = 1, and conductor N = 1.
- Degrees of L-functions: Degree-one Dirichlet L-functions can have larger conductor N, with coefficients given by multiplicative, periodic Dirichlet characters.
2.4. L-functions of degree 1.
Degree-one L-functions include Dirichlet L-functions, whose coefficients encode arithmetic information about primes in residue classes. Although degree one is completely classified, higher-degree L-functions are not.
- L-functions of degree 1: Dirichlet L-functions prove that infinitely many primes lie in each reduced arithmetic progression, with the N = 4 example also showing equal distribution between two classes.
- L-functions of degree 1: Degree one has a complete list of L-functions, whereas no complete classification exists for degrees greater than one.
- L-functions of degree 1: L-functions arise from diverse mathematical objects, including algebraic number fields and algebraic varieties such as curves.
- L-functions of degree 1: For many complicated objects, the L-function is defined but has not been proved to satisfy the defining L-function axioms.This was historically true even for elliptic curves over Q, while elliptic curves over real quadratic fields became known to be modular by 2013.
2.6. L-functions of number fields.
Every number field has a Dedekind zeta function with analytic properties analogous to those of the Riemann zeta-function. These properties connect algebraic prime factorization with analytic number theory, while GRH remains unresolved.
- L-functions of number fields: A number field is a finite extension of Q, and its Dedekind zeta function ζK(s) is defined similarly to ζ(s).
- L-functions of number fields: Analytic properties of ζK(s) yield information about prime factorizations in K.For cyclotomic fields Q(e2πi/m), combining algebraic and analytic properties of ζK(s) proves Dirichlet’s theorem on primes in arithmetic progressions.
- L-functions of number fields: The Generalized Riemann Hypothesis for Dedekind zeta functions ζK(s) remains unsolved.
2.7. L-functions of curves.
Curves over number fields produce L-functions whose degree depends on the field degree and curve genus, linking elliptic curves with modular forms. For degree four, important modularity connections remain partly conjectural and incomplete.
- L-functions of curves: An algebraic curve’s L-function degree depends on both the field degree and the curve’s genus.An elliptic curve over Q has degree 2, while an elliptic curve over a degree-d field has degree 2d.
- L-functions of curves: Degree-two L-functions are widely believed to come from products of degree-one L-functions, elliptic curves over Q, or special modular forms.
- L-functions of curves: For elliptic curves over Q, elliptic curves and modular forms produce the same L-functions through modularity.This connection underlies the theorem that every elliptic curve over Q is modular.
- L-functions of curves: For degrees 3 and 4, there is not yet a conjecture covering all sources of L-functions, and some expected connections remain unproved.
- L-functions of curves: Elliptic curves over real quadratic fields and Hilbert modular forms over the same field can produce the same degree-four L-functions.The cited modularity result establishes this connection for the real quadratic setting described.
- L-functions of curves: Over imaginary quadratic fields, elliptic-curve modularity is conjectured rather than proved in general, though individual curves can be handled by the Serre-Faltings-Livné method.The method uses Galois representations, and the database work applies it to prove modularity for its curves.
- L-functions of curves: Experiments and data collection remain important for degree-four L-functions because a general modularity proof over imaginary quadratic fields appears distant.
2.9. Showing connections through the LMFDB.
The LMFDB links mathematical objects sharing an L-function, while requiring careful consistency checks when independently computed data are joined.
- LMFDB links elliptic curves over real quadratic fields with Hilbert modular forms over the same field and their associated L-functions.
- Each elliptic-curve homepage links to its associated Hilbert modular form and L-function, with reverse links under development.
- Maintaining consistent object labels is a project difficulty when different contributors supply related datasets.
- For matching objects over Q(√5), contributors checked that independently generated Hilbert modular-form and elliptic-curve labels matched exactly.
3. The LMFDB database
The LMFDB combines a flexible database with a Python-based open-source software stack to store and serve a growing collection of mathematical data.
- MongoDB holds nearly a terabyte of data and indices, using a flexible schema that supports evolving mathematical records.PyMongo connects the database to Python tools including Flask and SageMath.
- Python lowers barriers for contributors while providing modules for database access, web serving, templating, and testing.
- The LMFDB stores around 35 databases containing mathematical objects and knowledge-database content.
- Records can gain additional fields later, and indexed values are stored when they are expensive to compute or useful for searches.
- The project uses off-the-shelf software with acknowledged imperfections and relies on contributors learning, reviewing, and testing the software.
- MongoDB cannot natively store integers larger than 2^32 or arbitrary rational numbers, so many such values are represented as strings.
4. The LMFDB website
The LMFDB website organizes mathematical data through object homepages, browsing and search tools, downloadable code, and expandable knowledge entries.
- The website presents, visualizes, browses, searches, documents, and provides downloadable access to LMFDB data.
- Serving several audiences simultaneously remains difficult, and the project lacks dedicated technical support staff.
- Every object has a permanent, mathematically meaningful URL, with pages generated from templates using stored and on-the-fly data.
- Related-object boxes connect pages for elliptic curves, number fields, and modular forms to associated L-functions.
- Object pages can provide code that recreates objects in SageMath, Pari/gp, or Magma for further work.
- Browse pages target users without underlying theory, whereas Search pages support experts querying labels or mathematical properties.
- Knowls expand glossary and knowledge content inside the page and can be dismissed without pop-ups or new pages.
5. The LMFDB project
The LMFDB is an internationally supported, consensus-driven collaboration that combines workshops, funding, technical infrastructure, and broad mathematical contributions.
- The project began at an AIM workshop in 2007 and uses regular workshops plus smaller focused groups to develop specific projects.
- Although Editorial and Management Boards exist, essentially all decisions are made by consensus at workshops.
- NSF and EPSRC grants have funded workshops, database and website servers, and some technical software support.
- The project requires contributors with diverse mathematical expertise, including collaborators who provide data rather than website code.