# L-FUNCTIONS & MODULAR FORMS DATABASE

| | homeAIMATH.ORG | AIMNEWS | 05.10.2016

|| EXPLORING THE MATHEMATICAL UNIVERSE ||

A team of more than 80 Mathematicians from 12 countries has begun charting the terrain of rich , new Mathematical worlds , and sharing their discoveries on the Web .

The Mathematical universe is filled with both familiar and exotic items , many of which are being made available for the first time .

The L-FUNCTIONS & MODULAR FORMS DATABASE , abbreviated LMFDB , is an intricate catalog of Mathematical objects and the connections between them .

Making those relationships visible has been made possible largely by the coordinated efforts of a group of researchers developing new algorithms and performing calculations on an extensive network of computers .

The project provides a new tool for several branches of Mathematics , Physics , and Computer Science .

Just like explorers , Mathematicians seek to discover paths between apparently unrelated areas .

Such discoveries can lead to breakthroughs when the connections are made explicit . . . .

The scale of the computational effort involved in the LMFDB is staggering : a total of nearly a thousand years of computer time spent on calculations by multiple teams of researchers .

A recent contribution by Andrew Sutherland at MIT used 72,000 cores of Google's Compute Engine to complete in one weekend a tabulation that would have taken more than a century on a single computer .

As noted by Sutherland , " computations in number theory are often amenable to parallelization , and this makes it easy to scale them to the cloud " .

The application of large-scale cloud computing to research in Pure Mathematics is just one of the ways in which the project is pushing forward the frontier of Mathematics .

MIT | NEWS | 05.10.2016

|| INTERNATIONAL TEAM LAUNCHES VAST ATLAS OF MATHEMATICAL OBJECTS ||

An international group of Mathematicians at MIT and other institutions has released a new online resource that provides detailed maps of previously uncharted Mathematical terrain .

The L-FUNCTIONS & MODULAR FORMS DATABASE , or LMFDB , is a detailed atlas of Mathematical objects that maps out the connections between them .

The LMFDB exposes deep relationships and provides a guide to previously uncharted territory that underlies current research in several branches of Physics , Computer Science , and Mathematics .

This coordinated effort is part of a massive collaboration of researchers around the globe .

LMFDB | L-FUNCTIONS & MODULAR FORMS DATABASE | INTRODUCTION

|| DATABASE OF L-FUNCTIONS , MODULAR FORMS , & RELATED OBJECTS ||

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 LMFDB universe 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 the LMFDB website will lead to the development and understanding of new Mathematics .

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