Power Savings for Counting (Twisted) Abelian Extensions of Number Fields

Abstract

We prove significant power savings for the error term when counting abelian extensions of number fields (as well as the twisted version of these results for nontrivial Galois modules). In some cases over $\mathbb{Q}$, these results reveal lower order terms following the same structure as the main term that were not previously known. Assuming the generalized Lindelöf hypothesis for Hecke $L$-functions, we prove square root power savings for the error compared to the order of the main term.

46 pages, v2 has been updated to include new references, correct various typos, and include a short appendix on alternate orderings and restricted local conditions