 

Artūras Dubickas
Nonreciprocal units in a number field with an application to OeljeklausToma manifolds view print


Published: 
March 18, 2014

Keywords: 
nonreciprocal unit, Pisot unit, OeljeklausToma manifold, LCK metric 
Subject: 
11R06, 11R21, 11R27, 53C55 


Abstract
In this paper we show that if a number field K contains
a nonreciprocal unit u of degree s+2t with s positive conjugates and
2t complex conjugates of equal moduli, where t ≧ 2, then s=(2t+2m)q2t for some integers m ≧ 0 and q ≧ 2.
On the other hand, for any s and t ≧ 2 related as above we construct a number field K with s real and 2t complex embeddings that contains a nonreciprocal unit u of degree s+2t with s positive conjugates and
2t complex conjugates of equal moduli. From this, for any pair of integers s ≧ 1,
t ≧ 2 satisfying s ≠ (2t+2m)q2t we deduce that the rank of the subgroup of units U whose 2t complex conjugates have equal moduli is smaller than s and, therefore, for any choice of an admissible subgroup A of K the corresponding OeljeklausToma manifold X(K,A) admits no locally conformal Kähler metric.


Acknowledgements
This research was supported by the Research Council of Lithuania grant No. MIP068/2013/LSS110000740


Author information
Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius LT03225, Lithuania
arturas.dubickas@mif.vu.lt

