# 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).

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.

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

### Organisations

- University of Warwick, United Kingdom (Lead Research Organisation)
- University of Luxembourg (Collaboration)
- University of Calgary, Canada (Collaboration)
- Ecole Polytechnique (Collaboration)
- University of Paris South 11, France (Project Partner)
- Northwestern University, United States (Project Partner)
- King's College London, United Kingdom (Fellow, Project Partner)

## People |
## ORCID iD |

Lassina Dembele (Principal Investigator / Fellow) |

### Publications

Berger T
(2015)

*Theta lifts of Bianchi modular forms and applications to paramodularity*in Journal of the London Mathematical Society
Billerey Nicolas
(2018)

*Some extensions of the modular method and Fermat equations of signature $(13,13,n)$*in arXiv e-prints
Breuil C
(2014)

*Sur un problème de compatibilité local-global modulo p pour GL2*in Journal für die reine und angewandte Mathematik (Crelles Journal)
Cremona John
(2019)

*On rational Bianchi newforms and abelian surfaces with quaternionic multiplication*in arXiv e-prints
Cunningham Clifton
(2017)

*Lifts of Hilbert modular forms and application to modularity of abelian varieties*in arXiv e-prints
Demb
(2019)

*Special hypergeometric motives and their $L$-functions: Asai recognition*in arXiv e-prints
Dembele Lassina
(2019)

*An intriguing hyperelliptic Shimura curve quotient of genus 16*in arXiv e-prints
Dembele Lassina
(2017)

*Compatibility between base change and Hecke orbits of Hilbert newforms*in arXiv e-prints
Dembélé L
(2014)

*On the computation of algebraic modular forms on compact inner forms of $\mathbf {GSp}_4$*in Mathematics of Computation
Dembélé L
(2015)

*Examples of abelian surfaces with everywhere good reduction*in Mathematische AnnalenDescription | 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 |