Explicit methods for algebraic automorphic forms

Lead Research Organisation: University of Warwick
Department Name: Mathematics

Abstract

The Langlands Programme is one of the most ambitious research programmes in modern mathematics. The programme is a vast web of conjectures which connect analysis, algebraic geometry, representation theory and number theory. At the heart of this programme are automorphic forms (analytic objects) and Galois representations (geometric objects). The two sets are related by a series of conjectures whose level of precision varies according to our understanding. The programme warrant our attention as it has far reaching consequences.

This project will develop explicit methods for algebraic automorphic forms. By providing concrete examples of automorphic to work with, it will help improve our understanding of many of conjectures in this field. The PI is particularly interested in the mod p Langlands, as he has had some success in this case with the group GL(2).

Planned Impact

Automorphic forms are arguably some of the most abstract objects in modern mathematics, and studying them by explicit methods is a nascent subject, except for classical modular forms. As explained in Part I of Previous Research and Track Record, the PI was the first person to develop practical algorithms for Hilbert modular forms. Although many of the current algorithms are efficient enough for academic purpose, there are still several major obstacles to overcome before they can leave that sphere. For that reason, the PI doesn't foresee any immediate non-academic benefit of this project. However he remains confident that this will change in the medium or long term.

For further details and explanations, please see the Pathways to Impact.

Publications

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Dembélé L (2020) An intriguing hyperelliptic Shimura curve quotient of genus 16 in Algebra & Number Theory

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Cremona J (2021) Arithmetic Geometry, Number Theory, and Computation

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Dembele Lassina (2017) Compatibility between base change and Hecke orbits of Hilbert newforms in arXiv e-prints

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Dembélé L (2013) Elliptic Curves, Hilbert Modular Forms and Galois Deformations

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Dembélé L (2015) Examples of abelian surfaces with everywhere good reduction in Mathematische Annalen

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Lassina Dembélé (2017) Inertial Langlands correspondence for $\GL_2$ and arithmetic applications

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Cunningham Clifton (2017) Lifts of Hilbert modular forms and application to modularity of abelian varieties in arXiv e-prints

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Lassina Dembélé (2019) Non-paritious Hilbert modular forms in Mathematische Zeitschrift

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Dembélé L (2014) On the computation of algebraic modular forms on compact inner forms of 4 in Mathematics of Computation

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Dembélé L (2021) On the existence of abelian surfaces with everywhere good reduction in Mathematics of Computation

 
Description 1. Found new examples of paramodular abelian surfaces using theta lifts of Hilbert and Bianchi modular forms.

2. Developed algorithms for computing partial weight one Hilbert modular forms.

3. Currently developing algorithms for automorphic forms on orthogonal and unitary groups.

4. Use lifts of Hilbert modular forms to prove modularity of certain abelian varieties. This provides the first substantial evidence to a conjecture of Dick Gross.

5. Prove the existence of non-paritious weight Hilbert modular forms, and use them to provide evidence for a conjecture of Buzzard-Gee on Galois representations.
Exploitation Route In the coming year(s), most of my time will be spending implementing the algorithms I have now developed. I will then use those algorithms to investigate new theoretical ideas, especially regarding automorphic forms on orthogonal/unitary groups as well as partial weight one Hilbert modular forms. I will also be working on developing explicit methods for inertial types and Langlands correspondence for GL(2).
Sectors Education
URL http://homepages.warwick.ac.uk/staff/L.Dembele/paper.html
 
Description Algebraic automorphic forms on orthogonal groups 
Organisation Ecole Polytechnique
Country France 
Sector Academic/University 
PI Contribution Develop algorithms for automorphic forms over compact orthogonal groups.
Collaborator Contribution Provide some of the theoretical tool need in order to understand automorphic forms on orthogonal groups. For example, my partner helped me better understand endoscopy and Arthur's multiplicities for such forms.
Impact This is still work in progress.
Start Year 2013
 
Description Lifts of Hilbert modular forms and applications to a conjecture of Gross 
Organisation University of Calgary
Department Department of Mathematics
Country Canada 
Sector Academic/University 
PI Contribution My collaborator and I have proved the existence of higher rank automorphic forms using Hilbert modular forms. We have used those lifts to prove case of a conjecture of Dick Gross on modularity of abelian varieties.
Collaborator Contribution See above
Impact Work in progress, we have nearly finished writing our first joint publication.
Start Year 2012
 
Description Partial weight one Hilbert modular forms 
Organisation University of Luxembourg
Department Mathematics Research Unit
Country Luxembourg 
Sector Academic/University 
PI Contribution Develop new algorithms for computing partial and parallel weight one Hilbert modular forms. These forms are very important for a better understanding of the local-global compatibility of the Langlands correspondence for GL(2). Our algorithm will provide the first tools for working with these forms concretely.
Collaborator Contribution See above
Impact I recently visited the University of Luxembourg to work on the implementation of our algorithm. We hope to have this completed by January 2015.
Start Year 2012