Quadratic p-ring spaces for counting dihedral fields

2014/03/16 by Daniel C. Mayer, Mayer, Daniel C. · 1 citation
Mathematics · #11R11 #11R16 #11R20 #11R29 #11Y40 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1403.3906

openalex publication_date 2014/03/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let p denote an odd prime. For all p-admissible conductors c over a quadratic number field \(K=ℚ(√(d))\), p-ring spaces \(V_p(c)\) modulo c are introduced by defining a morphism \(ψ: f↦ V_p(f)\) from the divisor lattice \(ℕ\) of positive integers to the lattice S of subspaces of the direct product \(V_p\) of the p-elementary class group \(C/C^p\) and unit group \(U/U^p\) of K. Their properties admit an exact count of all extension fields N over K, having the dihedral group of order 2p as absolute Galois group \(Gal(N | ℚ)\) and sharing a common discriminant \(d_N\) and conductor c over K. The number \(m_p(d,c)\) of these extensions is given by a formula in terms of positions of p-ring spaces in S, whose complexity increases with the dimension of the vector space \(V_p\) over the finite field \(\mathbbF_p\), called the modified p-class rank \(σ_p\) of K. Up to now, explicit multiplicity formulas for discriminants were known for quadratic fields with \(0≤σ_p≤ 1\) only. Here, the results are extended to \(σ_p=2\), underpinned by concrete numerical examples.

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