International Congress of Mathematicians

t.176 S. S. Kudlais the exponential integral. Recall that this function has a log singularity as r goesto zero and decays exponentially as r goes to infinity. In fact, as shown in [10],section 11, for any x £ V(M.) with Q(x) ^ 0, the function(,(x,z) :=ßi(2nR(x,z))can be viewed as a Green function on D for the divisor D x := {z £ D | (x, z) = 0}.A simple calculation, [10], shows that, for t > 0, E(t,v) is a Green function oflogarithmic type for the cycle Z(t), while, for t < 0, E(t,v) is a smooth function onM(C).Definition 2. (i) For t £ Z and v > 0, the class Z(t,v)£ CH (M) is defined by:Z(t,v)( (Z(t),E(t,v)) ift>0,^0! + (0,c^log(w)) ift = 0,{ (0,E(t,v)) ift p\D(B)where C,D(B)(S) = C,( s ) U P \D(B)(^ ^P S )anI(T) is a (nonholomorphic) modular„ —~~iform of weight |, valued in CH (A4), for a subgroup Y' c SL 2 (Z).The proof of Theorem 1 depends on Borcherd's result [1] and on the modularityof various complex valued g-expansions obtained by taking height pairings of