Abstract
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.
Highlights
Until the World Wide Web, such tables were hard to use, let alone to make, as they were only available in printed form, or on microfiche! An example relevant for the L-functions and Modular Forms Database (LMFDB) is the 1976 Antwerp IV tables of elliptic curves, published as part of a conference proceedings in Springer Lecture Notes in Mathematics 476, as a computer printout with manual amendments and diagrams
Some of the defining properties have not been proved for all the types of L-function in the database: this can be very hard! For example, Andrew Wiles proved Fermat’s Last Theorem by proving the modularity of certain elliptic curves over Q, which amounted to showing that the L-functions associated to elliptic curves really are L-functions in the above sense
It is widely believed that all degree 2 L-functions arise as follows: they either are products of two degree 1 L-functions, or come from elliptic curves over Q, or from modular form
Summary
Since the early days of using computers in number theory, computations and tables have played an important part in experimentation, for the purpose of formulating and proving (or disproving) conjectures. Until the World Wide Web, such tables were hard to use, let alone to make, as they were only available in printed form, or on microfiche! Even since the WWW, tables and databases have been scattered among a variety of personal web pages (including my own [1]). You had to know who to ask, download data, and deal with a wide variety of formats. In some areas of number theory, such as elliptic curves, the situation is much better and easier: packages such as SageMath [2], Magma [3], and Pari/gp [4] contain elliptic curve databases (sometimes as optional add-ons, as they are large).
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