Pointwise Bound for $\ell$-torsion of Class Groups

Abstract: $\ell$-torsion conjecture states that $\ell$-torsion of the class group $|\text{Cl}_K[\ell]|$ for every number field $K$ is bounded by $\text{Disc}(K)^{\epsilon}$. It follows from a classical result of Brauer-Siegel, or even earlier result of Minkowski that the class number $|\text{Cl}_K|$ of a number field $K$ are always bounded by $\text{Disc}(K)^{1/2+\epsilon}$, therefore we obtain a trivial bound $\text{Disc}(K)^{1/2+\epsilon}$ on $|\text{Cl}_K[\ell]|$. We will talk about recent works on breaking the trivial bound for $\ell$-torsion of class groups in some cases based on the work of Ellenberg-Venkatesh. We will also mention several questions following this line.

algebraic geometrynumber theory

Audience: researchers in the topic

UCSB Seminar on Geometry and Arithmetic