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Cycles on Siegel threefolds and derivatives of Eisenstein series. (English) Zbl 1045.11044
From the text: We consider the Siegel modular variety of genus 2 and a \(p\)-integral model of it for a good prime \(p>2\), which parametrizes principally polarized abelian varieties of dimension two with a level structure. We consider algebraic cycles on this model which are characterized by the existence of certain special endomorphisms and their intersections. We characterize that part of the intersection which consists of isolated points in characteristic \(p\) only. Furthermore, we relate the (naive) intersection multiplicities of the cycles at isolated points to special values of derivatives of certain Eisenstein series on the metaplectic group in 8 variables.
This paper is the first of a pair in which we generalize some of the results of S. Kudla [Ann. Math. (2) 146, 545–646 (1997; Zbl 0990.11032)] to higher dimensions in the case of finite primes of good reduction. In the companion paper S. Kudla and M. Rapoport [J. Reine Angew. Math. 515, 155–244 (1999; Zbl 1048.11048)] we are concerned with the Shimura variety associated to an orthogonal group of signature \((2,2)\) which is related to certain Hilbert-Blumenthal surfaces. We now give an overview of the structure of this paper. In Section 1, we introduce the Shimura variety and formulate the moduli problem solved by \({\mathcal M}\). Our special cycles are introduced in Section 2. We define the fundamental matrix in Section 3 and isolate there the part of \({\mathcal Z}\) lying purely in characteristic \(p\). It is clear from the above description that to proceed further we need a thorough understanding of the supersingular locus of \({\mathcal M}\times_{\text{Spec}\,\mathbb{Z}_{(p)}}\text{Spec}\,\mathbb{F}_p\). This is essentially due to L. Moret-Bailly and F. Oort. In Section 4, we give a presentation of their results in terms of Dieudonné theory, better suited for our needs. A similar presentation was independently given by C. Kaiser for a different purpose. The heart of the paper is Section 5. In it we determine the space of special endomorphisms of certain Diendonné modules and deduce the characterization of isolated intersection points (Theorems 5.11, 5.12 and 5.14). Here again the exceptional isomorphism plays a vital role. In Section 6, we explain the reduction of the calculation of \(e(\xi)\) to the result of B. H. Gross and K. Keating, and, in Section 7, we explain how to count the number of isolated points. Section 8 is a review of the Fourier coefficients of Siegel-Eisenstein Series. In Section 9, we bring everything together and prove the identity of the second main result (Theorem 0.2). In Section 10, we review some results of Y. Kitaoka and show how they can be used to prove the formulas on Whittaker functions needed in Section 9. Finally, there is an appendix containing some facts on Clifford algebras in our special situation.

MSC:
11G18 Arithmetic aspects of modular and Shimura varieties
11G50 Heights
11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)
11F30 Fourier coefficients of automorphic forms
14G35 Modular and Shimura varieties
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