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Groups of special units

  • Author(s): Beeson, Amanda M.;
  • et al.

Our work extends Anderson's construction of the maximal almost Abelian extension of the rational numbers and Kubert's proof that the Siegel group generates the full unit group up to 2-cotorsion. It is related to Sinnott's index calculations and relies heavily on the machinery of distributions developed by Kubert. In the cyclotomic setting, we prove the second order vanishing of a character combination of Hurwitz zeta functions and calculate the lead term. From this we derive a family of new trigonometric identities. Finally, we give a general algorithm for finding an explicit square root of a certain combinations of circular numbers that have a square root. In the imaginary quadratic setting, we give a combination of Siegel units that has a square root. We prove that if the square root of a modular unit has a level then that level is twice the level of the function itself.

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