2-Selmer Groups, 2-Class Groups, and the Arithmetic of Binary Forms

Explore a comprehensive lecture on 2-Selmer groups, 2-class groups, and the arithmetic of binary forms delivered by Ashvin A. Swaminathan from Princeton University at the Fields Institute. Delve into advanced number theory topics, including the definition and properties of R, fractional ideals, Hecke's theorem, orbit parametrization, and the parametrize-and-count strategy. Examine the 2-torsion in the class group of R, integral solutions to superelliptic equations, and 2-Selmer groups of hyperelliptic Jacobians. Gain insights into elliptic curves, their Selmer groups, and progress toward related conjectures. Investigate the second moment and class group applications in this 57-minute seminar, part of the Fields Number Theory Seminar series.

Intro
Roadmap
The Definition of R
Properties of R
Fractional Ideals of R
A Theorem of Hecke
Questions
Orbit Parametrization
Construction of An Integral Orbit
Image of the Parametrization
Primer on Parametrize-and-Count Strategy
2-Torsion in the Class Group of R
Integral Solutions to Superelliptic Equations (cont'd.)
2-Selmer Groups of Hyperelliptic Jacobians (cont'd.)
Varying the Leading Coefficient (cont'd.)
Error from Davenport's Lemma (cont'd.)
2-Selmer Groups of Genus-1 Curves
Background on Elliptic Curves and their Selmer groups
Progress toward the Conjecture
The Second Moment
Class Group Application