International Congress of Mathematicians

Eisenstein Series and Arithmetic Geometry 177with other classes in CH (A4). We now describe some of these in terms of valuesand derivatives of a certain Eisenstein series, [18], of weight §£I(T,S,D(B))= Y, (cT + d)-i\cT + d\- {s -i ) v^s-i )^i(sr/,D(B)),associated to F and the lattice F, and normalized so that it is invariant unders H> —s. The main result of joint work with M. Rapoport and T. Yang is thefollowing:Theorem 2. ([18]) (i)^I(T,—,D(B)) = deg(i(T)) =J2deg Q (Z(t,v))q t .(ii)£[(T,\;D(B))2'= (MT),û) = 5^(f (t, w ),w>g*.tNote that this result expresses the Fourier coefficients of the first two terms inthe Laurent expansion at the point s = \ of the Eisenstein series £I(T, S; D(Bj) interms of the geometry and the arithmetic geometry of cycles on A4.Next consider the image of-MrWi(r, i; F(F)) • degM- 1•cDin CH 1 (MQ), the usual Chow group of the generic fiber. By (i) of Theorem 2, itlies in the Mordell-Weil space CH 1 (Mq)° ® C ~ Jac(M)(Q) ® z C. In fact, it isessentially the generating function defined by Borcherds, [1], for the Shimura curveM, and hence is a holomorphic modular of weight §. For the case of modularcurves, such a modular generating function, whose coefficients are Heegner points,was introduced by Zagier, [25]. By the Hodge index theorem for CH (A4), [2], theproof of Theorem 1 is completed by showing that the pairing of