Lecture series of Dr. Artane Siad
Dr. Artane Siad
2024-08-30 14:30-16:00
MCM110
Speaker: Dr. Artane Siad (Princeton University)
Time: 14:30-16:00 August 26th (Monday), August 29th (Thursday), August 30th (Friday)
Venue: MCM110
Title 1: Average 2-torsion in the class group of monogenised fields
Abstract 1: The Cohen-Lenstra heuristics were introduced in the 1980s to explain the behaviour of the $p$-primary parts of the class groups of number fields varying over a fixed degree and signature. After reviewing these heuristics, I will show how to obtain an upper bound on the average number of $2$-torsion elements in the class group of monogenised fields of any degree $n \ge 3$. Conditional on a widely expected tail estimate, these upper bounds are equalities. Our result, which extends one of Bhargava-Hanke-Shankar in the cubic case, suggests that the Cohen-Lenstra heuristics fail for monogenic fields.
Title 2: Affine symmetric spaces and 2-torsion in the class group of unit-monogenized cubic fields
Abstract 2: Davenport's lemma is a key input to recent applications of the geometry-of-numbers to arithmetic statistics. Its absence in non-affine settings presents a major roadblock to progress on many natural questions. I will report on joint work with Arul Shankar and Ashvin Swaminathan, where we bound, and conditionally compute, the average number of $2$-torsion elements in the class group of unit-monogenised cubic fields. The proof substitutes homogeneous dynamics for Davenport's lemma in Bhargava's averaging method.
Title 3: Spin structures and quadratic enhancements on number fields
Abstract 3: I will report on joint work in progress with Akshay Venkatesh. Motivated by anomalous class group statistics, we propose an arithmetic analogue of the topological story of quadratic enhancements associated with spin structures on closed oriented 2- and 3-manifolds: a choice of spin structure gives, respectively, a quadratic refinement of the mod 2 intersection form and of the linking pairing on the first torsion homology. Time permitting, I will clarify the sense in which our construction can be viewed as a candidate determinant-of-cohomology for number fields.
