Arithmetic Hirzebruch Zagier cycles

S. Kudla; M. Rapoport

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arXiv · math/9904083

Arithmetic Hirzebruch Zagier cycles

S. Kudla·M. Rapoport

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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.

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S. Kudla, M. Rapoport. 1999-04-16. Arithmetic Hirzebruch Zagier cycles. https://arxiv.org/abs/math/9904083

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