Student Study Seminar on the Work of Manjul Bhargava

During this seminar, we are planning to give an introduction to Manjul Bhargava's work that made him a Field medalist. We will prove the theorems over density of discriminants of quartic and quintic fields, and the average size of Selmer groups. The seminar is for graduate students in the area of number theory, but since the techniques are very elemental, undergraduates interested in the topic could also get involved. Please email js6157@columbia.edu if you would like to join.

The seminar takes place in room 528 in the mathematics hall, from 5pm to 6pm every Friday. Also, see the notes I wrote by myself parts 1,2 and part 3.

A tentative schedule for the seminar will be as follows:

A tentative schedule for the seminar will be as follows:

Date | Speaker | Title and Abstract |
---|---|---|

09/08/2023 | Jiahe Shen | Introduction and composition laws induced by Bhargava cubes |

09/15/2023 | Jiahe Shen | Relations between composition laws with ideal classes in quadratic orders |

09/22/2023 | Jiahe Shen | Higher composition law II |

10/06/2023 | Jiahe Shen | Higher composition law III |

10/13/2023 | Jiahe Shen | Higher composition law IV |

10/27/2023 | Ajith Nair | The density of discriminants of quartic rings and fields (The first week) |

11/03/2023 | Jiahe Shen | The density of discriminants of quartic rings and fields (The second week) |

11/10/2023 | Jiahe Shen | The density of discriminants of quintic rings and fields |

11/17/2023 | Jiahe Shen | The average size of p-Selmer groups (Introduction and sketch proof for case p=2) |

11/21/2023 | Jiahe Shen | The average size of p-Selmer groups (p=2: Part I) |

12/01/2023 | Jiahe Shen | The average size of p-Selmer groups (p=2: Part II) |

12/04/2023 | Jiahe Shen | The average size of p-Selmer groups (p=3,5) |

12/11/2023 | Jiahe Shen | Applications to the BSD rank conjecture |

References

[1]M. Bhargava, Higher composition laws, Ph.D. Thesis, Princeton University, June 2001.

[2]M. Bhargava, Higher composition laws. I. A new view on Gauss composition, and quadratic generalizations, Annals of Mathematics(2004), 217-250.

[3]M. Bhargava, Higher composition laws. II. On cubic analogues of Gauss composition, Annals of Mathematics(2004), 865-886.

[4]M. Bhargava, Higher composition laws. III. The parametrization of quartic rings, Annals of Mathematics(2004), 1329-1360.

[5]M. Bhargava, Higher composition laws. IV. The parametrization of quintic rings, Annals of Mathematics(2008), 53-94.

[6]M. Bhargava, The density of discriminants of quartic rings and fields, Annals of Mathematics(2005), 1031-1063.

[7]M. Bhargava, The density of discriminants of quintic rings and fields, Annals of Mathematics(2010), 1559-1591.

[8]M. Bhargava and S. Arul, Binary quartic forms having bounded invariants, and the boundness of average rank of elliptic curves, Annals of Mathematics(2015), 191-242.

[9]M. Bhargava and S. Arul, Ternary cubic forms have bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0, Annals of Mathematics(2015), 587-621.

[10]M. Bhargava and S. Arul, The average size of the 5-selmer group of the elliptic curves is 6, and the average rank is less than 1, December 2013.

[11]M. Bhargava, C. Skinner, and W. Zhang, A majority of elliptic curves over Q satisfy the Birch and Swinnerton-Dyer conjecture, July 2014.