LMF: L-Functions and Modular Forms

Lead Research Organisation: University of Warwick
Department Name: Mathematics

Abstract

L-functions and modular forms are fundamental mathematical objects that encode much of our knowledge of contemporary number theory. They form part of a web of interconnected objects, the understanding of which in the most basic cases lies at the foundations of much of modern mathematics. A spectacular example is Wiles' proof of Fermat's Last Theorem, which was an application of a fundamental "modularity" link between L-functions, modular forms and elliptic curves. This project will greatly extend and generalize such connections, both theoretically and computationally.

The research vision inspiring our programme can be summarised as: "Breaking the boundaries of classical L-functions and modular forms, and exploring their applications to 21st-century mathematics, physics, and computer science". Our guiding goal is to push forward both theoretical and algorithmic developments, in order to develop L-functions and modular forms far beyond current capabilities. This programme will systematically develop an extensive catalogue of number theoretic objects, and will make this information available through an integrated online resource that will become an indispensable tool for the world's research community.

L-functions are to pure mathematics what fundamental particles are to physics: their interaction reveal fundamental truths. To continue the analogy, computers are to number theorists what colliders are to particle physicists. Aside from their established role as empirical "testers" for conjectures and theories, experiments can often throw up quite unexpected phenomena which go on to reshape modern theory. Our programme will establish a major database and encyclopedia of knowledge about L-functions and related objects, which will play a role analogous to that of the LHC for the scientists at CERN. Both are at the threshold of tantalising glimpses into completely uncharted territories: higher degree L-functions for us and the Higgs boson for them.

Theoretical and computational work on higher degree L-functions has only started to make substantial progress in the past few years. There do not currently exist efficient methods to work with these, and rigorous computations with them are not yet possible. Neither is there yet an explicit description of all ways in which degree 3 L-functions can arise. We will address these facets in our research programme: both algorithmic development and theoretical classification.

As well as having theoretical applications to modularity relationships as in Wiles' proof, detailed knowledge of L-functions has more far-reaching implications. Collections of L-functions have statistical properties which first arose in theoretical physics. This surprising connection, which has witnessed substantial developments led by researchers in Bristol, has fundamental predictive power in number theory; the synergy will be vastly extended in this programme. In another strand, number theory plays an increasingly vital role in computing and communications, as evidenced by its striking applications to both cryptography and coding theory.

The Riemann Hypothesis (one of the Clay Mathematics Million Dollar Millennium Problems) concerns the distribution of prime numbers, and the correctness of the best algorithms for testing large prime numbers depend on the truth of a generalised version of this 150-year-old unsolved problem. These are algorithms which are used by public-key cryptosystems that everyone who uses the Internet relies on daily, and that underpin our digital economy. Our programme involves the creation of a huge amount of data about a wide range of modular forms and L-functions, which will far surpass in range and depth anything computed before in this area. This in turn will be used to analyse some of the most famous outstanding problems in mathematics, including the Riemann Hypothesis and another Clay problem, the Birch and Swinnerton-Dyer conjecture.

Planned Impact

Those to benefit from the planned research certainly include mathematicians (especially number theorists), physicists and computer scientists, as well as a wider range of scientists who need to manage large amounts of data.

UK universities will benefit through the raised international profile of its mathematics research, through greater engagement of the public in its research, and through the attraction of well-qualified international students to study in their thriving, world-class research institutes.

The UK economy will benefit from the increased number of skilled mathematicians who are not only experts in their specialised field, but who have spent three years on a project where the presentation of their results to potential users has been key. Many of these are likely to continue to academic careers, while others will be highly sought-after for posts in the digital and knowledge economy, having acquired transferable skills in database management, website design and the visualisation of complex sets of data.

Economic and societal benefits will follow (in due course) from a better understanding by physicists of complex quantum systems (all modern electronic systems depend on quantum physics), and this understanding will be assisted by the study of L-functions since they exhibit all of the characteristic features of the spectra of complex quantum systems, yet are substantially more amenable to numerical modelling.

Additional societal benefits will come from applications of our work in cryptography, which is vital for the security and reliability of the internet, digital commerce, and national security.

One of the most spectacular results of the last 25 years, which made front-page news worldwide, was the proof of Fermat's Last Theorem through greater structural understanding of L-functions, which are the objects of study in our proposal. Three of the ten Fields medals awarded so far in this century were for work closely connected to the our L-function database, and we expect that applications of higher degree L-functions will similarly produce some of the greatest results of 21st century mathematics.
 
Description Horizon 2020
Amount € 7,643,071 (EUR)
Funding ID 676541 
Organisation European Commission H2020 
Sector Public
Country Belgium
Start 09/2015 
End 08/2019
 
Description Marie Curie Fellowship--Galois Representations and Diophantine Problems--Nuno Freitas
Amount € 183,454 (EUR)
Funding ID 747808 
Organisation European Commission H2020 
Sector Public
Country Belgium
Start 03/2018 
End 02/2020
 
Description Marie Sklodowska-Curie Individual Fellowships
Amount € 195,454 (EUR)
Funding ID 793646 - LowDegModCurve - H2020-MSCA-IF-2017 
Organisation European Commission 
Sector Public
Country European Union (EU)
Start 09/2018 
End 08/2020
 
Title LMFDB website 
Description A website allowing other researchers easy access to research data produced by the project and contributed by third parties. 
Type Of Material Improvements to research infrastructure 
Year Produced 2013 
Provided To Others? Yes  
Impact Around 150 publications cite the website for use of the data: http://www.lmfdb.org/citation/citations 
URL http://www.lmfdb.org
 
Title L-functions and modular forms database 
Description A large collection of data containing elliptic curves over Q and over number fields, L functions of degrees 1, 2, 3 and 4, modular forms (classical, Hilbert, Bianchi, Siegel, Maass), local and global number fields, and zeros of L-functions. 
Type Of Material Database/Collection of data 
Year Produced 2013 
Provided To Others? Yes  
Impact Ongoing 
URL http://www.lmfdb.org
 
Description Simons Collaboration on Arithmetic Geometry, Number Theory, and Computation 
Organisation Brown University
Country United States 
Sector Academic/University 
PI Contribution I am an affiliated scientist on this major US collaboration because of my leadership on the LMF project.
Collaborator Contribution By hiring several postdoctoral researchers and running workshops, this collaboration will continue the work currently being carried out by the researchers on the EPSRC project LMF into the next decade.
Impact None yet
Start Year 2017
 
Title eclib 
Description C++ library for elliptic curves and modular symbols 
Type Of Technology Software 
Year Produced 2016 
Open Source License? Yes  
Impact Key component to Sagemath opensource mathematics system. Responsible for computation of the database of elliptic curves over the rationals. 
URL https://www.zenodo.org/record/58276
 
Description April 2017 Min Lee Numerical Computations with the Selberg trace formula Seminar, KAIST, Daejeon, Republic of Korea 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Other audiences
Results and Impact April 2017 Min Lee Numerical Computations with the Selberg trace formula Seminar, KAIST, Daejeon, Republic of Korea
Year(s) Of Engagement Activity 2017
 
Description April 2017 Min Lee Numerical Computations with the Selberg trace formula Seminar, POSTECH, Pohang, Republic of Korea 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Other audiences
Results and Impact April 2017 Min Lee Numerical Computations with the Selberg trace formula Seminar, POSTECH, Pohang, Republic of Korea
Year(s) Of Engagement Activity 2017
 
Description January 2017 Min Lee Numerical Computations with the Selberg trace formula JMM MAA Invited paper session on L-functions and other animals, Atlanta GA, USA 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Other audiences
Results and Impact January 2017 Min Lee Numerical Computations with the Selberg trace formula JMM MAA Invited paper session on L-functions and other animals, Atlanta GA, USA
Year(s) Of Engagement Activity 2017
 
Description June 2016 Min Lee Effective equidistribution of rational points on expanding horospheres 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Other audiences
Results and Impact June 2016 Min Lee Effective equidistribution of rational points on expanding horospheres
Seminar, King's College London, UK
Year(s) Of Engagement Activity 2016
 
Description November 2016 Min Lee Effective equidistribution of rational points on expanding horospheres 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Other audiences
Results and Impact November 2016 Min Lee Effective equidistribution of rational points on expanding horospheres
Seminar, University of Oxford, UK
Year(s) Of Engagement Activity 2016
 
Description November 2017 Min Lee Numerical Computations with the Selberg trace formula Seminar, University of Nottingham, UK 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Other audiences
Results and Impact November 2017 Min Lee Numerical Computations with the Selberg trace formula Seminar, University of Nottingham, UK
Year(s) Of Engagement Activity 2017
 
Description September 2017 Min Lee Effective equidistribution of rational points on expanding horospheres 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Other audiences
Results and Impact September2017 Min Lee Effective equidistribution of rational points on expanding horospheres
Automorphic forms and arithmetic workshop, Oberwolfach, Germany
Year(s) Of Engagement Activity 2017
 
Description • December 2015, Algebra Seminar, Bar Ilan University. 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Other audiences
Results and Impact • December 2015, Algebra Seminar, Bar Ilan University.
Year(s) Of Engagement Activity 2015
 
Description • December 2015, Algebra Seminar, Haifa University. 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Other audiences
Results and Impact • December 2015, Algebra Seminar, Haifa University.
Year(s) Of Engagement Activity 2015
 
Description • December 2015, Dynamics Seminar, Technion - Israel Institute of Technology. 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Other audiences
Results and Impact • December 2015, Dynamics Seminar, Technion - Israel Institute of Technology.
Year(s) Of Engagement Activity 2015
 
Description • December 2016, Bar-Ilan University. 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Other audiences
Results and Impact • December 2016, Bar-Ilan University. Colloquium.
Year(s) Of Engagement Activity 2016
 
Description • December 2016, Haifa University. 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Other audiences
Results and Impact • December 2016, Haifa University. Colloquium
Year(s) Of Engagement Activity 2016
 
Description • December 2016, Number Theory Seminar, Tel Aviv University. 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Other audiences
Results and Impact • December 2016, Number Theory Seminar, Tel Aviv University.
Year(s) Of Engagement Activity 2016
 
Description • December 2016, Number Theory Seminar, The Hebrew University. 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Other audiences
Results and Impact • December 2016, Number Theory Seminar, The Hebrew University.
Year(s) Of Engagement Activity 2016
 
Description • December 2016, special talk, Ben Gurion University. 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Other audiences
Results and Impact • December 2016, special talk, Ben Gurion University.
Year(s) Of Engagement Activity 2016
 
Description • December 2017, Algebra Seminar, Technion - Israel Institute of Technology. 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Other audiences
Results and Impact • December 2017, Algebra Seminar, Technion - Israel Institute of Technology.
Year(s) Of Engagement Activity 2017
 
Description • February 2015, Number Theory Seminar, University of Bristol. 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Other audiences
Results and Impact • February 2015, Number Theory Seminar, University of Bristol.
Year(s) Of Engagement Activity 2015
 
Description • February 2018, London Number Theory Seminar, UCL. 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Other audiences
Results and Impact • February 2018, London Number Theory Seminar, UCL.
Year(s) Of Engagement Activity 2018
 
Description • January 2016, Number Theory and Function Fields at the Crossroads, University of Exeter, UK. 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Other audiences
Results and Impact • January 2016, Number Theory and Function Fields at the Crossroads, University of Exeter, UK.
Year(s) Of Engagement Activity 2016
 
Description • January 2017, Joint Mathematics Meetings, Session: L-functions and other animals, Atlanta, Georgia. 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Other audiences
Results and Impact • January 2017, Joint Mathematics Meetings, Session: L-functions and other animals, Atlanta, Georgia.
Year(s) Of Engagement Activity 2017
 
Description • July 2015, Galois Theory and Number Theory, University of Konstanz, Germany. 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Other audiences
Results and Impact • July 2015, Galois Theory and Number Theory, University of Konstanz, Germany.
Year(s) Of Engagement Activity 2015
 
Description • March 2015, Number Theory Seminar, Tel Aviv University. 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Other audiences
Results and Impact • March 2015, Number Theory Seminar, Tel Aviv University.
Year(s) Of Engagement Activity 2015
 
Description • March 2016, British Mathematical Colloquium, Number theory workshop, University of Bristol, UK. 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Other audiences
Results and Impact • March 2016, British Mathematical Colloquium, Number theory workshop, University of Bristol, UK.
Year(s) Of Engagement Activity 2016
 
Description • March 2017, Number Theory Seminar, Exeter University. 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Other audiences
Results and Impact • March 2017, Number Theory Seminar, Exeter University.
Year(s) Of Engagement Activity 2017
 
Description • November 2017, Number Theory Seminar, University of Manchester. 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Other audiences
Results and Impact • November 2017, Number Theory Seminar, University of Manchester.
Year(s) Of Engagement Activity 2017