Arithmetic Hirzebruch Zagier cycles

Abstract

We define special cycles on arithmetic models of twisted Hilbert-Blumenthal surfaces at primes of good reduction. These are arithmetic versions of these cycles. In particular, we characterize the non-degenerate intersections and partially determine the generating series formed from the intersection numbers of them relating it to the value at the center of symmetry of the derivative of a certain metaplectic Eisenstein series in 6 variables. These results are analogous to those obtained by us in the case of Siegel threefolds (alg-geom/9711025). We also study the case of degenerate intersections and show that in this case the intersection locus is a configuration of projective lines whose dual graph is described in terms of subcomplexes of the Bruhat-Tits building of PGL(2,F), where F is an unramified quadratic extension of Q_p.