Some Applications of Kronecker's Limit Formulas

Divisibility of anticyclotomic L-functions and theta functions with complex multiplication

“…Now the first assertion follows, since the Siegel functions g ab take integral values at points in imaginary quadratic fields, and take units as values for suitable parameters a and b [Ra,p. 127].…”

Construction of Ray class fields by elliptic units

“…From complex multiplication we know that elliptic units are contained in certain ray class fields over a quadratic imaginary number field K, and Ramachandra [3] has shown that these ray class fields can even be generated by elliptic units. However the generators constructed by Ramachandra We will explain later, why our method of proof does not carry over to a full proof of the conjecture.…”

Class invariants and cyclotomic unit groups from special values of modular units

“…, . When is not a prime power, there may be multiplicative dependence between the elements in lim τ →0 S C , although these primitive cyclotomic units in Q(ζ ) + may be used to define a set of multiplicatively independent units as given by Ramachandra [21]. This proves Theorem 1.2..…”