Power Savings When Counting Abelian Extensions of Number Fields
Let $K$ be a number field and $A$ an abelian group. We prove power savings for the number of $A$-extensions of $K$ with discriminant bounded above by $X$, as $X$ tends towards infinity. This includes proving the existence of secondary terms following an analogous structure to the main term proposed by Malle and proven by Wright. We will sketch the proof, highlighting an extension of the local Tate pairing and the explicit meromorphic continuation of a large family of Euler products. We also discuss the generalization of these results to power savings for the "twisted" version of Malle's conjecture for the number of coclasses in $H^1(K,T)$ with bounded discriminant, where $T$ is a finite Galois module.