Torsion in class groups reading group
Recall that to each number field, one can associate a finite group known as its ideal class group. This group can measure the failure of unique factorisation in rings of integers, and is connected to methods for solving Diophantine equations. In this term we will study bounds for the torsion of class groups, a topic that involves integral points on curves, and has applications to ranks of elliptic curves and counting Galois extensions. Alternatively, one can regard this topic as a unifying thread, bringing together methods that would be otherwise unrelated.
We will study the methods of the following four papers.
- Pierce, A bound for the 3-part of class numbers of quadratic fields by means of the square sieve.
- Helfgott--Venkatesh, Integral points on elliptic curves and 3-torsion in class groups.
- Ellenberg--Venkatesh, Reflection principles and bounds for class group torsion.
- Heath-Brown--Pierce, Averages and moments related to class numbers of imaginary quadratic fields.
The reading group will take place at 1pm on Fridays in B3.03.
| Week |
Speaker |
Title |
Notes |
|
2 |
Joseph | Introductory talk | Latex notes |
| 3 | Akshat | Pierce (I) | |
| 4 | Rubin | Pierce (II) | Handwritten notes |
| 5 | Joe | Canonical and local heights, Tate parameterisation | Handwritten notes |
| 6 | Harry | Helfgott--Venkatesh (I) | Latex notes |
| 7 | Owen | Helfgott--Venkatesh (II) | Latex notes |
| 8 | Seth | Ellenberg--Venkatesh | |
| 9 | Sam | Heath-Brown--Pierce | |
| 10 |