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arxiv: 2602.23619 · v2 · pith:V2L63YB3new · submitted 2026-02-27 · 🧮 math.NT

Counting number fields using multiple Dirichlet series

Brandon Alberts , Alina Bucur This is my paper

Pith reviewed 2026-05-25 06:47 UTC · model grok-4.3

classification 🧮 math.NT
keywords number fieldsGalois groupsDirichlet seriescounting asymptoticsinertial invariantsnilpotent groupsasymptotic formulas
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The pith

Multiple Dirichlet series count number fields with fixed Galois groups ordered by inertial invariants.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a technique using multiple Dirichlet series to count number fields with a given Galois group. This produces unconditional asymptotic counts for infinitely many new groups that previous methods could not handle. Conditional on subconvexity for the series, it establishes growth rates for further families, such as all groups of nilpotency class two, and improves error terms for nilpotent groups.

Core claim

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. Conditional on subconvexity bounds, asymptotic growth rates hold for infinitely many new groups G where minimum index elements lie in unions of proper abelian normal subgroups, including all nilpotency class 2 groups, and power saving errors apply when G is nilpotent.

What carries the argument

Multiple Dirichlet series encoding the inertial invariants of the Galois extensions.

If this is right

  • Unconditional counting results hold for infinitely many new concentrated and semiconcentrated groups.
  • Conditional asymptotic growth rates exist for groups with minimum index elements in unions of abelian normal subgroups.
  • All groups of nilpotency class 2 admit conditional asymptotics.
  • Nilpotent groups receive power saving error terms in the counts.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This analytic method could be adapted to count extensions ordered by other arithmetic invariants.
  • Improved bounds on Dirichlet series growth would convert more conditional results to unconditional ones.
  • The results suggest that similar series constructions may resolve counting problems for additional classes of Galois groups.

Load-bearing premise

Subconvexity bounds hold for the multiple Dirichlet series associated to the Galois groups.

What would settle it

Numerical evidence that the number of G-extensions for a nilpotency class 2 group grows at a different rate than predicted would disprove the conditional claim.

Figures

Figures reproduced from arXiv: 2602.23619 by Alina Bucur, Brandon Alberts.

Figure 1
Figure 1. Figure 1: Region of Absolute Convergence Our goal is to construct a meromorphic continuation to a larger segment of these lines, which we will do by constructing a meromorphic continuation of the entire multiple Dirichlet series. Each of the three orderings we are considering for D4 is semiconcentrated in the abelian normal subgroups TB :“ x2A, 2By “ 1A Y 2A Y 2B and TC :“ x2A, 2Cy “ 1A Y 2A Y 2C. (In fact, the quar… view at source ↗
Figure 2
Figure 2. Figure 2: Continuation in TC-direction We can already see that some progress has been made for the quartic discriminant. Indeed, a meromorphic continuation constructed from a single abelian normal subgroup in this way is another way of thinking about the inductive methods of [ALOWW25] for concentrated groups. The dashed line represents a possible pole at s2C “ 1 for DQpD4; sq in this region, which would be canceled … view at source ↗
Figure 3
Figure 3. Figure 3: Continuation in TB- and TC-directions σ2B σ2C -1 0 1 2 3 -1 0 1 2 3 conductor, octic quartic s2B “ 1 s2C “ 1 [PITH_FULL_IMAGE:figures/full_fig_p026_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Continuation to the convex hull have shown that DQpD4; sq is analytic on this region except for possible poles at s2A “ 1, s2B “ 1, and s2C “ 1 that could have been canceled out by ppsq. It takes significant extra work to determine the orders of these poles, especially at the intersection points like pσ2B, σ2Cq “ p1, 1q. Equation (2.9) lets us interpolate the vertical bounds between ΩB and ΩC. By plugging … view at source ↗
Figure 5
Figure 5. Figure 5: Projection of UB Y UC Y UD onto the pσ2B, σ2C, σ2Dq “ px, y, zq space From the image, it is clear to see that the convex hull of this region is the orthant σ2B, σ2C, σ2D ą 1{2 with the edges and corner sliced off. We can compute these by com￾puting the convex hull between the various corners of UB, UC, and UD (these are the outer corners in [PITH_FULL_IMAGE:figures/full_fig_p060_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Continuation in TC-direction One can see that this complex line appears to cross both the polar divisors s2B “ 1 and s2C “ 1 within the region of meromorphicity, suggesting that we might reveal a secondary term [PITH_FULL_IMAGE:figures/full_fig_p070_6.png] view at source ↗
Figure 6
Figure 6. Figure 6: We expect this include a polar divisor at [PITH_FULL_IMAGE:figures/full_fig_p072_6.png] view at source ↗
read the original 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 Lindel\"of 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper develops a method based on multiple Dirichlet series to count number fields with fixed Galois group G, ordered by arbitrary inertial invariants. It establishes unconditional asymptotic formulas for infinitely many new concentrated and semiconcentrated groups not reachable by prior techniques. Conditionally on subconvexity bounds (such as the generalized Lindelöf hypothesis) for certain Dirichlet series, it obtains asymptotics for infinitely many additional groups whose minimal-index elements lie in a union of proper abelian normal subgroups, including all groups of nilpotency class 2. When G is nilpotent the method yields a power-saving error term.

Significance. If the analytic derivations and error-term estimates hold, the work meaningfully enlarges the set of Galois groups admitting counting theorems, both unconditionally and under standard subconvexity hypotheses. The separation of unconditional results for concentrated/semiconcentrated groups from conditional results for a broader class (including all class-2 nilpotents) is a clear strength, as is the power-saving error term for nilpotent G. The approach via multiple Dirichlet series supplies a new technical route that avoids some limitations of earlier methods.

minor comments (2)
  1. Abstract, line 3: the phrase 'subconvexity bounds bounds' contains a duplicated word; this should be corrected for clarity.
  2. The manuscript would benefit from an explicit statement, early in the introduction, of the precise inertial invariants used to order the extensions, together with a short comparison table listing which groups fall into the unconditional, conditional, and previously known categories.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and significance assessment of our manuscript, as well as the recommendation for minor revision. No specific major comments appear in the report, so we have no individual points to address point-by-point. We remain available to incorporate any minor editorial or presentational changes the editor or referee may suggest in a revised version.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external analytic methods

full rationale

The paper applies established techniques from the study of multiple Dirichlet series to count G-extensions ordered by inertial invariants. Unconditional results are claimed for concentrated and semiconcentrated groups, while conditional results explicitly invoke external subconvexity bounds (e.g., generalized Lindelöf hypothesis) as a hypothesis rather than deriving them internally. No self-definitional equations, fitted inputs renamed as predictions, load-bearing self-citations, or ansatzes smuggled via prior work appear in the stated claims or abstract. The power-saving error term for nilpotent G is presented as a consequence of the method under the stated conditions, with no reduction to the paper's own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no information on free parameters, axioms, or invented entities; full text would be required to audit the analytic setup.

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Reference graph

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