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

Brandon Alberts

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.

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 , 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 -functions, we prove square root power savings for the error compared to the order of the main term.
Paper Structure (27 sections, 32 theorems, 196 equations)

This paper contains 27 sections, 32 theorems, 196 equations.

Table of Contents

  1. Introduction
  2. Power Saving Bounds for Counting Abelian Extensions
  3. Lower Order Terms
  4. Power Saving Bounds for Counting Twisted Abelian Extensions
  5. History of Previous Results
  6. Layout of the Paper
  7. Preliminaries
  8. Summary of Alberts--O'Dorney
  9. Twisted Permutation Representation
  10. Explicit Meromorphic Continuation
  11. Parametrizing the Local Tate Pairings at Tame Places
  12. Constructing the Galois representation
  13. A Power Saving Tauberian Theorem
  14. The Difference Operators
  15. Shifting the Contour Integral
  16. ...and 12 more sections

Key Result

Theorem 1.2

Let $K$ be a number field and $G$ an abelian group. The generating Dirichlet series for $G$-extensions of $K$ ordered by discriminant is meromorphic on the right halfplane ${\rm Re}(s) > 0$. Any poles with real part larger than $\frac{1}{2a(G)}$ occur at $s=\frac{1}{d}$ for each $d\in \textnormal{in Here, $\chi:G_K\to \hat{Z}^\times$ is the cyclotomic character and induces a Galois action on $G$ b

Theorems & Definitions (63)

  • Remark 1.1
  • Theorem 1.2
  • Definition 1.3
  • Remark 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Conjecture 1.8
  • Corollary 1.9
  • Corollary 1.10
  • ...and 53 more