Final Report Summary - CBGA (Cohomology of Bianchi Groups and Arithmetic)
Executive Summary:
Holding a prestigious and very generous Marie Curie Intra-European Fellowship at a top university like the University of Warwick has given an immense boost to fellow's career: he has produced first-rate research (three papers already submitted, one paper available on the internet, four papers halfway done), established and developed many fruitful collaborations, extended his mathematical horizons significantly, promoted his research all around Europe and North America, and as the crowning achievement, he has obtained a permanent faculty position at the University of Sheffield, a first rate university in the UK with a strong group in the fellow's research area.
Summary of The Original Proposal:
The cohomology of arithmetic groups sit at the crossroads of many parts of mathematics, including algebraic topology, algebraic and differential geometry, representation theory, and number theory. It is conjoined with the theory of automorphic forms and, via the Langlands Programme, provides a way to understand the mysteries of the absolute Galois group of the rationals.
The role in the Langlands programme of torsion classes in the cohomology of arithmetic groups is far from being well understood, and is one of the central problems in number theory today. The scientific goal of this proposal was to gain insight on this problem by studying it in one of the simplest and oldest settings; namely, in the case of Bianchi groups.
The overall research goal of this proposal was to understand better the nature of torsion in the homology of Bianchi groups. More specifically, (PROJECT 1) the relationship between
the torsion with even mod p Galois representations of the absolute Galois group of the field of rational numbers, and (PROJECT 2) with genus 2 cohomological type Siegel modular forms.
On the training side, the project had specific training goals aiming to gain (and advance existing) skills and a working knowledge in computational number theory,
arithmetic of Siegel modular forms, p-adic modular forms and p-adic L-functions, the Jacquet-Langlands correspondence and arithmetic aspects of Dynamical Systems.
Research Outcomes:
During his Marie Curie Fellowship at the University of Warwick, the fellow has been very productive. He has finished four projects (the papers associated to these projects have been submitted to mathematics journals of solid international reputation) and has several on-going ones some of which are close to finishing.
Many of the components of PROJECT 1 (see above) have been completed. An important first step was the determination of the dimension of the spaces of Bianchi modular forms of non-trivial levels that were lifted from classical (elliptic) modular forms. He has accomplished this task with the collaboration of Dr. Panagiotis Tsaknias (Luxembourg). In fact, the results obtained in the first step were of independent interest and therefore they decided to publish them as a separate publication. The write-up is available on the internet and will be submitted to a suitable journal very soon. For the second step, the fellow has collected a good amount of data with the computer programs that he wrote. However in order to significantly extend the scope of the data, he needs to extend his programs to treat the cohomology of Bianchi groups with non-trivial Nebentypus, which is currently underway.
During the first year of the fellowship, Dr. Lassina Dembele and the fellow worked out the necessary theoretical background and laid down a detailed plan of action for PROJECT 2. The fellow has written the necessary computer programs and has carried out the first step of the project, which was to determine suitable mod p Hecke eigenvalues systems which do not arise as mod p reductions of Hecke eigenvalue systems associated to Bianchi modular forms. For the second step of the project Dr.Dembele has been working on extending his computer programs. They expect to collect data by the end of the summer of 2014. It should be mentioned that the discussions of the fellow and Dr.Dembele in the first year of the fellowship lead to another project which they completed jointly with Tobias Berger (Sheffield) and Ariel Pacetti (Buenos Aires). This paper the arose from this project has already been submitted.
During his fellowship, the fellow has initiated many additional projects some of which are already completed. Due to lack of space we shall only briefly list them here:
PROJECT 3: Torsion homology growth and cycle complexity of arithmetic manifolds, with N.Bergeron (Paris 6) and A.Venkatesh (Stanford) (completed)
PROJECT 4: Stark-Heegner points for elliptic curves over number fields of arbitrary signature, with X.Guitart (Essen) and M.Masdeu (Columbia) (completed)
PROJECT 5: Lifts of Bianchi Modular Forms and Applications to the Paramaodularity Conjecture, with T.Berger (Sheffield), L.Dembele (Warwick), A.Pacetti (Buenos Aires) (completed)
PROJECT 6: Dimension Formulae for Spaces of Lifted Modular Forms for GL(2), with P.Tsaknias (completed)
PROJECT 7: Bianchi Modular Forms Database within the LMFDB, with J.Cremona (Warwick), A.Rahm (Galway), D.Yasaki (North Carolina), A.Page (Bordeaux)
PROJECT 8 : Arithmetic Kleinian groups and the integral Jacquet-Langlands correspondence, with A.Page (Bordeaux)
PROJECT 9: Noncommutative Geometry and homology of arithmetic manifolds, B.Mesland (Warwick)
PROJECT 10: Transfer Operator for Bianchi Groups, with M.Fraczek (Warwick)
The fellow also acquired very valuable training during his fellowship. Topics include
TRAINING 1: Advanced skills in the non-mathematical side of computational number theory
TRAINING 2 : Jacquet-Langlands correspondence and its application
TRAINING 3: Arithmetic of Siegel modular forms of genus 2
TRAINING 4: Dynamical Systems and their connections to number theory
TRAINING 5: Noncommutative Geometry and its connections to number theory
Moreover, during his first year at Warwick, the fellow has gained valuable experience in teaching and mentoring. In Term 1 of the academic year 2012-13, he had full responsibility for lecturing and assessing a masters-level course on ``Modular Forms" which was attended by a group of 28 students, including 3rd and 4th year undergraduates, 1st year Masters and PhD students. He received very positive evaluations from the students about his performance as a teacher. He also supervised a 4th year undergraduate masters level project on Galois Theory. To further his skills in teaching and mentoring, he voluntarily enrolled in the "PCAPP" (Postgraduate Certificate in Academic and Professional Practice) programme of the University of Warwick. This is a university wide programme designed for newly joined permanent faculty.
Holding a prestigious and very generous Marie Curie Intra-European Fellowship at a top university like the University of Warwick has given an immense boost to fellow's career: he has produced first-rate research (three papers already submitted, one paper available on the internet, four papers halfway done), established and developed many fruitful collaborations, extended his mathematical horizons significantly, promoted his research all around Europe and North America, and as the crowning achievement, he has obtained a permanent faculty position at the University of Sheffield, a first rate university in the UK with a strong group in the fellow's research area.
Summary of The Original Proposal:
The cohomology of arithmetic groups sit at the crossroads of many parts of mathematics, including algebraic topology, algebraic and differential geometry, representation theory, and number theory. It is conjoined with the theory of automorphic forms and, via the Langlands Programme, provides a way to understand the mysteries of the absolute Galois group of the rationals.
The role in the Langlands programme of torsion classes in the cohomology of arithmetic groups is far from being well understood, and is one of the central problems in number theory today. The scientific goal of this proposal was to gain insight on this problem by studying it in one of the simplest and oldest settings; namely, in the case of Bianchi groups.
The overall research goal of this proposal was to understand better the nature of torsion in the homology of Bianchi groups. More specifically, (PROJECT 1) the relationship between
the torsion with even mod p Galois representations of the absolute Galois group of the field of rational numbers, and (PROJECT 2) with genus 2 cohomological type Siegel modular forms.
On the training side, the project had specific training goals aiming to gain (and advance existing) skills and a working knowledge in computational number theory,
arithmetic of Siegel modular forms, p-adic modular forms and p-adic L-functions, the Jacquet-Langlands correspondence and arithmetic aspects of Dynamical Systems.
Research Outcomes:
During his Marie Curie Fellowship at the University of Warwick, the fellow has been very productive. He has finished four projects (the papers associated to these projects have been submitted to mathematics journals of solid international reputation) and has several on-going ones some of which are close to finishing.
Many of the components of PROJECT 1 (see above) have been completed. An important first step was the determination of the dimension of the spaces of Bianchi modular forms of non-trivial levels that were lifted from classical (elliptic) modular forms. He has accomplished this task with the collaboration of Dr. Panagiotis Tsaknias (Luxembourg). In fact, the results obtained in the first step were of independent interest and therefore they decided to publish them as a separate publication. The write-up is available on the internet and will be submitted to a suitable journal very soon. For the second step, the fellow has collected a good amount of data with the computer programs that he wrote. However in order to significantly extend the scope of the data, he needs to extend his programs to treat the cohomology of Bianchi groups with non-trivial Nebentypus, which is currently underway.
During the first year of the fellowship, Dr. Lassina Dembele and the fellow worked out the necessary theoretical background and laid down a detailed plan of action for PROJECT 2. The fellow has written the necessary computer programs and has carried out the first step of the project, which was to determine suitable mod p Hecke eigenvalues systems which do not arise as mod p reductions of Hecke eigenvalue systems associated to Bianchi modular forms. For the second step of the project Dr.Dembele has been working on extending his computer programs. They expect to collect data by the end of the summer of 2014. It should be mentioned that the discussions of the fellow and Dr.Dembele in the first year of the fellowship lead to another project which they completed jointly with Tobias Berger (Sheffield) and Ariel Pacetti (Buenos Aires). This paper the arose from this project has already been submitted.
During his fellowship, the fellow has initiated many additional projects some of which are already completed. Due to lack of space we shall only briefly list them here:
PROJECT 3: Torsion homology growth and cycle complexity of arithmetic manifolds, with N.Bergeron (Paris 6) and A.Venkatesh (Stanford) (completed)
PROJECT 4: Stark-Heegner points for elliptic curves over number fields of arbitrary signature, with X.Guitart (Essen) and M.Masdeu (Columbia) (completed)
PROJECT 5: Lifts of Bianchi Modular Forms and Applications to the Paramaodularity Conjecture, with T.Berger (Sheffield), L.Dembele (Warwick), A.Pacetti (Buenos Aires) (completed)
PROJECT 6: Dimension Formulae for Spaces of Lifted Modular Forms for GL(2), with P.Tsaknias (completed)
PROJECT 7: Bianchi Modular Forms Database within the LMFDB, with J.Cremona (Warwick), A.Rahm (Galway), D.Yasaki (North Carolina), A.Page (Bordeaux)
PROJECT 8 : Arithmetic Kleinian groups and the integral Jacquet-Langlands correspondence, with A.Page (Bordeaux)
PROJECT 9: Noncommutative Geometry and homology of arithmetic manifolds, B.Mesland (Warwick)
PROJECT 10: Transfer Operator for Bianchi Groups, with M.Fraczek (Warwick)
The fellow also acquired very valuable training during his fellowship. Topics include
TRAINING 1: Advanced skills in the non-mathematical side of computational number theory
TRAINING 2 : Jacquet-Langlands correspondence and its application
TRAINING 3: Arithmetic of Siegel modular forms of genus 2
TRAINING 4: Dynamical Systems and their connections to number theory
TRAINING 5: Noncommutative Geometry and its connections to number theory
Moreover, during his first year at Warwick, the fellow has gained valuable experience in teaching and mentoring. In Term 1 of the academic year 2012-13, he had full responsibility for lecturing and assessing a masters-level course on ``Modular Forms" which was attended by a group of 28 students, including 3rd and 4th year undergraduates, 1st year Masters and PhD students. He received very positive evaluations from the students about his performance as a teacher. He also supervised a 4th year undergraduate masters level project on Galois Theory. To further his skills in teaching and mentoring, he voluntarily enrolled in the "PCAPP" (Postgraduate Certificate in Academic and Professional Practice) programme of the University of Warwick. This is a university wide programme designed for newly joined permanent faculty.