The distribution of the multiplicative index of algebraic numbers over residue classes

Pieter Moree; Antonella Perucca; Pietro Sgobba

doi:10.1007/s12188-024-00276-2

The distribution of the multiplicative index of algebraic numbers over residue classes

Pieter Moree
, Antonella Perucca
, Pietro Sgobba^*

^*Corresponding author for this work

Max Planck Institute for Mathematics
University of Luxembourg

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

Abstract

Let K be a number field and G a finitely generated torsion-free subgroup of K^×. Given a prime p of K we denote by ind_p(G) the index of the subgroup (Gmodp) of the multiplicative group of the residue field at p. Under the Generalized Riemann Hypothesis we determine the natural density of primes of K for which this index is in a prescribed set S and has prescribed Frobenius in a finite Galois extension F of K. We study in detail the natural density in case S is an arithmetic progression, in particular its positivity.

Original language	English
Pages (from-to)	1-17
Number of pages	17
Journal	Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg
Volume	94
Issue number	1
DOIs	https://doi.org/10.1007/s12188-024-00276-2
Publication status	Published - Apr 2024

Keywords

Multiplicative index and order
Natural density
Primary: 11R45
Primes in arithmetic progression
Reductions of algebraic numbers
Secondary: 11A07, 11R44

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10.1007/s12188-024-00276-2

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N2 - Let K be a number field and G a finitely generated torsion-free subgroup of K×. Given a prime p of K we denote by indp(G) the index of the subgroup (Gmodp) of the multiplicative group of the residue field at p. Under the Generalized Riemann Hypothesis we determine the natural density of primes of K for which this index is in a prescribed set S and has prescribed Frobenius in a finite Galois extension F of K. We study in detail the natural density in case S is an arithmetic progression, in particular its positivity.

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KW - Multiplicative index and order

KW - Natural density

KW - Primary: 11R45

KW - Primes in arithmetic progression

KW - Reductions of algebraic numbers

KW - Secondary: 11A07, 11R44

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JO - Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg

JF - Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg

IS - 1

ER -

The distribution of the multiplicative index of algebraic numbers over residue classes

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