Distribution of genus numbers of abelian number fields
Christopher Frei, Daniel Loughran, Rachel Newton
TL;DR
This paper investigates the distribution of genus numbers in abelian number field extensions, providing an asymptotic average and demonstrating the rarity of specific genus numbers.
Contribution
It offers a new asymptotic formula for the average genus number and proves that any fixed genus number occurs with zero density.
Findings
Derived an asymptotic formula for average genus numbers
Proved that any fixed genus number appears only 0% of the time
Enhanced understanding of genus number distribution in abelian extensions
Abstract
We study the quantitative behaviour of genus numbers of abelian extensions of number fields with given Galois group. We prove an asymptotic formula for the average value of the genus number and show that any given genus number appears only 0% of the time.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Analytic Number Theory Research · Cryptography and Residue Arithmetic