# Cohen-Lenstra heuristics, Brauer relations, and low-dimensional manifolds

Lead Research Organisation:
University of Glasgow

Department Name: School of Mathematics & Statistics

### Abstract

A concept of central importance in mathematics is that of symmetry. One used to think of symmetry as a property of geometric shapes, but in the 19th century Evariste Galois extended the concept of symmetry to algebraic objects, and today his insights are completely fundamental to pure mathematics. The underlying goal of this proposal, which is situated between Algebra, Number Theory, and Topology, relying also on techniques from Probability Theory and Additive Combinatorics, is to study symmetries of arithmetic and geometric objects.

Number Theory is an ancient mathematical discipline with a rich history of over 2000 years, but also with spectacular developments in recent years. Some of the most impressive recent advances have happened in the area of Number Theory called Arithmetic Statistics: the groundbreaking contributions of Manjul Bhargava have been rewarded with a Fields Medal in 2014. The aim of Arithmetic Statistics is to understand the behaviour of arithmetic objects, such as (ray) class groups, in families. The birth of this area goes back to Gauss, who formulated some concrete conjectures concerning the behaviour of class groups of quadratic fields. It was given a huge boost in the 1980s, when Cohen and Lenstra proposed a general model that implied all the conjectures of Gauss, and more. Roughly speaking, they postulated that class groups of imaginary quadratic fields obey a probability distribution that assigns to a finite abelian group X a probability that is inverse proportional to the number of symmetries of X. This is, in fact, a very natural model for random algebraic objects. This was later generalised to other number fields by Cohen and Martinet, but in more general cases the probability distributions looked more mysterious. The Cohen-Lenstra-Martinet Heuristics have been used as a guiding principle in Arithmetic Statistics since then, and have found applications in many other areas, such as the theory of Elliptic Curves, in Combinatorics, and in Differential Geometry. This project will consist of a blend of theorems, conjectures, and computations. I will:

- show that the original conjectures are false, as stated,

- find the correct formulations,

- put them on a more conceptual footing, by explaining the mysterious looking probability weights of Cohen-Martinet using a theory of commensurability of algebraic objects that I have been developing together with Hendrik Lenstra,

- extend the scope of the heuristics, e.g. to ray class groups.

Two other kinds of very basic objects whose symmetries one studies are finite sets and finite dimensional vector spaces. An old problem in Representation Theory, with applications to Number Theory and Differential Geometry, is to compare symmetries of sets with symmetries of vector spaces, and in particular to determine which symmetries of sets become isomorphic (essentially the same) when the sets are turned into vector spaces. There are two incarnations of this problem: one where the vector spaces are over a field of characteristic 0, e.g. over the real numbers, and one where they are vector spaces over a field of positive characteristic. In previous joint work with Tim Dokchitser we have solved the case of characteristic 0, thereby settling an over 60 year old problem. Using the techniques that we developed, and new ones, this project will settle the case of positive characteristic.

Finally, I will also investigate symmetries of low-dimensional manifolds. These are the basic objects studied by modern geometry and topology, and it is an old and fruitful line of investigation to determine what one can say about the topology of the manifold from knowing its symmetries. In recent joint work with Aurel Page, I have introduced a new representation theoretic tool into the area, which I had worked on in number theoretic contexts. Using these new techniques, I am planning to shed more light on the connection between symmetries and the topology of the manifold.

Number Theory is an ancient mathematical discipline with a rich history of over 2000 years, but also with spectacular developments in recent years. Some of the most impressive recent advances have happened in the area of Number Theory called Arithmetic Statistics: the groundbreaking contributions of Manjul Bhargava have been rewarded with a Fields Medal in 2014. The aim of Arithmetic Statistics is to understand the behaviour of arithmetic objects, such as (ray) class groups, in families. The birth of this area goes back to Gauss, who formulated some concrete conjectures concerning the behaviour of class groups of quadratic fields. It was given a huge boost in the 1980s, when Cohen and Lenstra proposed a general model that implied all the conjectures of Gauss, and more. Roughly speaking, they postulated that class groups of imaginary quadratic fields obey a probability distribution that assigns to a finite abelian group X a probability that is inverse proportional to the number of symmetries of X. This is, in fact, a very natural model for random algebraic objects. This was later generalised to other number fields by Cohen and Martinet, but in more general cases the probability distributions looked more mysterious. The Cohen-Lenstra-Martinet Heuristics have been used as a guiding principle in Arithmetic Statistics since then, and have found applications in many other areas, such as the theory of Elliptic Curves, in Combinatorics, and in Differential Geometry. This project will consist of a blend of theorems, conjectures, and computations. I will:

- show that the original conjectures are false, as stated,

- find the correct formulations,

- put them on a more conceptual footing, by explaining the mysterious looking probability weights of Cohen-Martinet using a theory of commensurability of algebraic objects that I have been developing together with Hendrik Lenstra,

- extend the scope of the heuristics, e.g. to ray class groups.

Two other kinds of very basic objects whose symmetries one studies are finite sets and finite dimensional vector spaces. An old problem in Representation Theory, with applications to Number Theory and Differential Geometry, is to compare symmetries of sets with symmetries of vector spaces, and in particular to determine which symmetries of sets become isomorphic (essentially the same) when the sets are turned into vector spaces. There are two incarnations of this problem: one where the vector spaces are over a field of characteristic 0, e.g. over the real numbers, and one where they are vector spaces over a field of positive characteristic. In previous joint work with Tim Dokchitser we have solved the case of characteristic 0, thereby settling an over 60 year old problem. Using the techniques that we developed, and new ones, this project will settle the case of positive characteristic.

Finally, I will also investigate symmetries of low-dimensional manifolds. These are the basic objects studied by modern geometry and topology, and it is an old and fruitful line of investigation to determine what one can say about the topology of the manifold from knowing its symmetries. In recent joint work with Aurel Page, I have introduced a new representation theoretic tool into the area, which I had worked on in number theoretic contexts. Using these new techniques, I am planning to shed more light on the connection between symmetries and the topology of the manifold.

### Planned Impact

As is often the case in pure mathematics, the main foreseeable impact of the proposed project will be within mathematics. Experience shows that while pure mathematics research can have a huge long term impact outside academia, it is difficult to foresee its nature at an early stage. I will therefore focus on accelerating the first step in the life cycle of scientific discovery from basic research to societal impact: academic dissemination of results, with the aim to reach a wide audience.

The channels of dissemination will include the traditional method of publication in appropriate journals, but also making preprints publicly available on the arXiv, on the insitutional preprint server, and on my personal homepage, which will greatly accelerate dissemination. I will also continue to give talks about my work at national and international conferences, workshops, colloquia and seminars, and will continue maintaining personal contact with researchers who I think may be interested in my results. The experimental data that I will obtain, and that will be of use to other researchers who want to understand the number theoretic objects that this project aims to understand (and have been forming the focus of number theoretic research for about two centuries) will also be made publicly available, both in the framework of already widely used databases, such as the EPSRC-funded LMFDB project, and on my personal website. The algorithms that I will develop will continue being incorporated into widely used scientific software, such as MAGMA.

There will also be direct non-academic impact of my work, through my outreach activities. I will continue visiting schools and explaining advanced mathematical topics, e.g. the Birch and Swinnerton-Dyer conjecture, to 6th form students. I will also continue giving public lectures in the department, both in and outside of the framework of open days, or as a plenary speaker at student run conferences. In the past, topics for these talks have included Dynamical Systems, Galois module structures, elliptic curves, and more. If time permits and if satisfactory arrangements with a school can be found, I am also hoping to revive my practice of offering longer running weekly mathematics/logic workshops for younger children, as I have done several times in the past.

The channels of dissemination will include the traditional method of publication in appropriate journals, but also making preprints publicly available on the arXiv, on the insitutional preprint server, and on my personal homepage, which will greatly accelerate dissemination. I will also continue to give talks about my work at national and international conferences, workshops, colloquia and seminars, and will continue maintaining personal contact with researchers who I think may be interested in my results. The experimental data that I will obtain, and that will be of use to other researchers who want to understand the number theoretic objects that this project aims to understand (and have been forming the focus of number theoretic research for about two centuries) will also be made publicly available, both in the framework of already widely used databases, such as the EPSRC-funded LMFDB project, and on my personal website. The algorithms that I will develop will continue being incorporated into widely used scientific software, such as MAGMA.

There will also be direct non-academic impact of my work, through my outreach activities. I will continue visiting schools and explaining advanced mathematical topics, e.g. the Birch and Swinnerton-Dyer conjecture, to 6th form students. I will also continue giving public lectures in the department, both in and outside of the framework of open days, or as a plenary speaker at student run conferences. In the past, topics for these talks have included Dynamical Systems, Galois module structures, elliptic curves, and more. If time permits and if satisfactory arrangements with a school can be found, I am also hoping to revive my practice of offering longer running weekly mathematics/logic workshops for younger children, as I have done several times in the past.

### Organisations

- University of Glasgow, United Kingdom (Fellow, Lead Research Organisation)
- The National Institute for Research in Computer Science and Control (INRIA) (Collaboration)
- Dartmouth College, United States (Project Partner)
- University of Wisconsin Madison, United States (Project Partner)
- University of Leiden, Netherlands (Project Partner)

### Publications

Bartel A
(2017)

*A note on Green functors with inflation*in Journal of Algebra
Bartel A
(2017)

*Commensurability of automorphism groups*in Compositio MathematicaDescription | One key finding has been the disproof, in joint work with Hendrik W. Lenstra Jr., of a 30 year old conjecture, the Cohen--Lenstra--Martinet heuristic. The heuristic aims to predict and explain the statistical behaviour of certain objects of fundamental importance in number theory, so-called ideal class groups. In my work with Lenstra, we have demonstrated that these objects have more structure to them than had hitherto been realised, and the models need to account for this structure. We have proposed corrected models, which also shed more light on why the observed behaviour is plausible. A separate key finding, in joint work with Aurel Page, is that certain representation theoretic techniques that had been developed in the context of elliptic curves can be adapted to a vastly more general situation, and for example used to study the question "What can one tell about the shape of a drum from the way it sounds". The work on exploring the consequences of this generalisation is still ongoing. |

Exploitation Route | I envisage wide ranging applications of the findings by number theorists, geometers and representation theorists in the study of statistical questions. A currently more distant dream is that the techniques that Page and I are developing might be used to tackle some major open problems in the theory of automorphic forms. |

Sectors | Other |

Description | Aurel Page |

Organisation | The National Institute for Research in Computer Science and Control (INRIA) |

Department | Bordeaux |

Country | France |

Sector | Public |

PI Contribution | Several months of research time by the PI; office space for visitor |

Collaborator Contribution | Several months of research time by collaborator; office space for PI when visiting the collaborator |

Impact | The collaboration is still ongoing, and some results are to appear, while others are still work in progress. |

Start Year | 2018 |

Description | Outreach talks |

Form Of Engagement Activity | Participation in an open day or visit at my research institution |

Part Of Official Scheme? | No |

Geographic Reach | National |

Primary Audience | Public/other audiences |

Results and Impact | Mathematical interactive presentations to school children who are about to enter university. |

Year(s) Of Engagement Activity | 2017,2018,2019 |

Description | Talks to undergraduates |

Form Of Engagement Activity | A talk or presentation |

Part Of Official Scheme? | No |

Geographic Reach | National |

Primary Audience | Undergraduate students |

Results and Impact | Talks to undergraduates from all over the UK on the major open problems and recent progress in algebraic number theory; one of the talks was as a plenary speaker at a student run conference (the YRM), another one was as a plenary speaker at an LMS summer school, and a third as an invited speaker at an undergraduate maths society meeting. |

Year(s) Of Engagement Activity | 2017,2018,2019 |