Number/Representation Theory

Let $K \neq \mathbb{Q}$ be a number field of degree $[K:\mathbb{Q}]$ and absolute discriminant $D_K = |\mathrm{Disc}(K)|$. Let $\mathrm{Cl}_K$ be the class group of $K$. For an integer $\ell \geq 2$, the $\ell$-torsion of the class group of $K$ satisfies the well-known trivial bound $$ |\mathrm{Cl}_K[\ell]| \leq |\mathrm{Cl}_K| \ll_{[K:\mathbb{Q}]} D_K^{1/2} (\log D_K)^{[K:\mathbb{Q}]-1} $$ due to Landau. Improvements over this trivial bound, both conditional and unconditional, have generated significant interest in many cases depending on $\ell$, the degree $[K:\mathbb{Q}]$, and the subfield structure of $K$. In this talk, I will discuss an unconditional log-power savings improvement over this trivial bound for all $\ell$ and all number fields $K$.

This is joint work with Robert Lemke Oliver.