On the p-rank of class groups of p-extensions

2022/12/19 by Yuan Liu, Liu, Yuan
Mathematics · #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2212.09888

openalex publication_date 2022/12/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove a local-global principle for the embedding problems of global fields with restricted ramification. By this local-global principle, for a global field k, we use only the local information to give a presentation of the maximal pro-p Galois group of k with restricted ramification, when some Galois cohomological conditions are satisfied. For a Galois p-extension K/k, we use our presentation result for k to study the structure of pro-p Galois groups of K. Then for k=ℚ and k=\mathbbF_q(t) with p\nmid q, we give upper and lower bounds for the rank of p-torsion group of the class group of K, and these bounds depend only on the structure of the Galois group and the inertia subgroups of K/k. Finally, we study the p-rank of class groups of cyclic p-extensions of ℚ and the 2-rank of class groups of multiquadratic extensions of ℚ, for a fixed ramification type.

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