Effective equidistribution of norm one elements in CM-fields

2025/07/14 by Akhtari, Shabnam, Vaaler, Jeffrey D., Widmer, Martin
#11D45 #11G50 #11R06 #11R21 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2507.10387

Abstract

For a number field K let S_K be the maximal subgroup of the multiplicative group K^× that embeds into the unit circle under each embedding of K into the complex numbers. The group S_K can be seen as an archimedean counterpart to the group of units O_K^× of the ring of integers O_K. If K=ℚ(S_K) is a CM-field then S_K/\mathop\rm Tor\nolimits(K^×) is a free abelian group of infinite rank. If K=ℚ(S_K) is not a CM-field then S_K=\± 1\. In the former case S_K is the kernel of the relative norm map from K^× to the multiplicative subgroup k^× of the maximal totally real subfield k of K.

Citations

• Equidistribution of Algebraic Numbers of Norm One in Quadratic Number\n Fields
• Manin's conjecture for toric varieties
• Hauteurs et mesures de Tamagawa sur les variétés de Fano

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