Modular functions and the uniform distribution of CM points

Abstract

(1) zd = { i √ d 2 if d ≡ 0 (mod 4), −1+i √ d 2 if d ≡ 3 (mod 4). The j-function has the remarkable property that j(zd) is an algebraic integer of degree h(−d), the class number of K = Q(zd). In fact, K(j(zd)) is the Hilbert class field of K [4]. The first few values of j(zd) are: j(z3) = 0, j(z4) = 12 , j(z7) = −153, j(z8) = 20, j(z11) = −323 j(z15) = −191025−85995√5 2 , j(z19) = −963, j(z20) = 632000+282880 √ 5,