{"id":"hexagon/2610.00502","pdf":"https://hexagonmath.org/pdf/2610.00502v1","url":"https://hexagonmath.org/abs/2610.00502v1","tags":["msc:11R45","Malle's conjecture","deformation theory","embedding problems"],"score":0,"title":"Galois embedding problems and number field counting","domain":"math.NT","source":"hexagon","authors":["Anwesh Ray"],"reviews":0,"subject":"math.NT","summary":"Let $p$ be a prime number and $1\\to H\\to G\\xrightarrow{q}T\\to1$ be an exact sequence of finite groups, where $H$ is a nontrivial $p$-group. Fix a continuous surjection $\\pi:\\GQ\\twoheadrightarrow T$ and finitely many $H$-conjugation-invariant local conditions $\\cD$. Let $N_{G,\\pi,\\cD}(X)$ count the continuous surjections $\\rho:\\GQ\\twoheadrightarrow G$ satisfying $q\\circ\\rho=\\pi$ and these local conditions, with $|\\Delta_{F_\\rho}|\\leq X$, where $F_\\rho=\\overline{\\Q}^{\\ker\\rho}$. If at least one such lift exists, we prove that, for every $\\varepsilon>0$, \\[ X^{p/(|G|(p-1))} \\ll_{G,\\pi,\\cD}…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-23","reactions":{},"withdrawn":false,"authorship":"human","formalized":false,"human_understanding":"all"}