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Galois embedding problems and number field counting

Primarily human-written textHuman understanding: all partsmath.NT — Number Theory

Contributed by Anwesh Ray ↗

Version 1 / Oct 23, 2026 / CC BY 4.0

Abstract

Let pp be a prime number and 1→H→G→qT→11\to H\to G\xrightarrow{q}T\to1 be an exact sequence of finite groups, where HH is a nontrivial pp-group. Fix a continuous surjection $\pi:\GQ\twoheadrightarrow T$ and finitely many HH-conjugation-invariant local conditions $\cD$. Let $N_{G,\pi,\cD}(X)$ count the continuous surjections $\rho:\GQ\twoheadrightarrow G$ satisfying q∘ρ=πq\circ\rho=\pi and these local conditions, with ∣ΔFρ∣≤X|\Delta_{F_\rho}|\leq X, where $F_\rho=\overline{\Q}^{\ker\rho}$. If at least one such lift exists, we prove that, for every ε>0\varepsilon>0, \[ X^{p/(|G|(p-1))} \ll_{G,\pi,\cD} N_{G,\pi,\cD}(X) \ll_{G,\pi,\cD,\varepsilon} X^{p/(|G|(p-1))+\varepsilon} \] for all sufficiently large XX. This establishes the power of XX predicted by Alberts's conjecture for every finite pp-group kernel, without any solvability assumption on TT. If every element of order pp in HH lies in Z(H)Z(H), then every nonempty family satisfies the asymptotic predicted by Alberts: \[ N_{G,\pi,\cD}(X) \sim C_{G,\pi,\cD} X^{p/(|G|(p-1))}(\log X)^{b_V-1}, \] with $C_{G,\pi,\cD}>0$. As an application, for odd pp and m≥2m\geq2, we obtain discriminant asymptotics for surjective lifts of a fixed residual Galois representation to $\GL_d(\Z/p^m\Z)$, subject to finitely many local conditions, whenever such a lift exists.

Provenance statement

The author used ChatGPT 6 Astra as an exploratory tool to study related papers and to assist in correcting mistakes in the article. It was not used to generate the mathematical ideas or the main results of the article.

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Version history

  1. v1Submitted by Anwesh RayInitial depositCurrentOct 23, 2026