UC San Diego

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.