Modular functions and the uniform distribution of CM points

Modular functions and the uniform distribution of CM points

(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) ...

Traces of Cm Values of Modular Functions

Zagier proved that the traces of singular moduli, i.e., the sums of the values of the classical j-invariant over quadratic irrationalities, are the Fourier coefficients of a modular form of weight 3/2 with poles at the cusps. Using the theta correspondence, we generalize this result to traces of CM values of (weakly holomorphic) modular functions on modular curves of arbitrary genus. We also st...

CM points and weight 3 / 2 modular forms

We survey the results of [Fun02] and of our joint work with Bruinier [BF06] on using the theta correspondence for the dual pair SL(2)×O(1, 2) to realize generating series of values of modular functions on a modular curve as (non)-holomorphic modular forms of weight 3/2.

Traces of Cm Values of Modular Functions and Related Topics

The purpose of this note is to report on recent joint work with J. Funke, P. Jenkins, and K. Ono on the traces of CM values of modular functions and some applications [BF], [BJO].

Nontriviality of Rankin-Selberg L-functions and CM points