Counting number fields using multiple Dirichlet series

Abstract

We provide a method for counting number fields of fixed Galois group ordered by arbitrary inertial invariants using analytic techniques from the study of multiple Dirichlet series. We prove unconditional results for infinitely many new (concentrated and semiconcentrated) groups that were not approachable by previous methods. Conditional on subconvexity bounds bounds for certain Dirichlet series (e.g. the generalized Lindelöf hypothesis), we use these techniques to prove the existence of an asymptotic growth rate for $G$-extensions for infinitely many new groups $G$ for which the minimum index elements of $G$ are contained in a union of proper abelian normal subgroups. In particular, our conditional results include all groups with nilpotency class $2$. Additionally, when $G$ is nilpotent our results give a power saving error term.

Figures (6)

• Figure 1: Region of Absolute Convergence
• Figure 2: Continuation in $T_C$-direction
• Figure 3: Continuation in $T_B$- and $T_C$-directions
• Figure 4: Continuation to the convex hull
• Figure 5: Projection of $U_B\cup U_C \cup U_D$ onto the $(\sigma_{2B},\sigma_{2C},\sigma_{2D}) = (x,y,z)$ space

Theorems & Definitions (107)

• Conjecture 1: Number Field Counting Conjecture
• Definition 1.1
• Lemma 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Definition 1.9