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Arithmetic Hirzebruch Zagier cycles. (English) Zbl 1048.11048

From the introduction: In this paper and its companion [S. Kudla and M. Rapoport [Ann. Sci. Éc. Norm. Supér., IV. Sér. 33, No. 5, 695–756 (2000; Zbl 1045.11044)], cited as [*] in the sequel, we establish a relation between part of the height pairing of special cycles on a Shimura variety associated to an orthogonal group of signature \((n-1,2)\) and special values of derivatives of Fourier coefficients of certain Siegel Eisenstein series of genus \(n\) in two new cases. Our results provide some evidence in favor of the general program set forth in [S. Kudla, Ann. Math. (2) 146, 545–646 (1997; Zbl 0990.11032)], where the case of Shimura curves was analyzed. In [*], we considered the case of (twisted) Siegel threefolds \((n=4)\) while, in the present paper, we are concerned with the case of (twisted) Hilbert-Blumenthal surfaces \((n=3)\).
For these surfaces, the special cycles include the modular and Shimura curves studied extensively by Hirzebruch and Zagier. Among other things, they showed that a generating function for the intersection numbers of such curves is an elliptic modular form of weight 2. The Hirzebruch-Zagier curves also account for all Tate classes on such surfaces rational over Abelian extensions of \(\mathbb{Q}\). Thus it is of interest to investigate the arithmetic analogues of these cycles on integral models of such surfaces, which are arithmetic threefolds.
The canonical model \(M\) over \(\mathbb{Q}\) of a Hilbert-Blumenthal surface is defined as a moduli space of Abelian varieties with real multiplications, and the Hirzebruch-Zagier curves on \(M\) can be defined as the loci where the Abelian variety admits an extra endomorphism of a particular type, a special endomorphism. For a prime \(p\) of good reduction, a model \({\mathcal M}\) of \(M\) over \(\mathbb{Z}_{(p)}\) can be defined by a moduli problem. We then consider a modular extension \({\mathcal Z}\) to \({\mathcal M}\) of a Hirzebruch-Zagier curve \(Z\) on \(M\), again defined by imposing a special endomorphism. These are the arithmetic Hirzebruch-Zagier cycles of the title. More generally, one can consider the loci where the Abelian variety carries several special endomorphisms. If two independent endomorphisms are imposed one obtains points on the generic fiber \(M\) and curves on \({\mathcal M}\). If three independent endomorphisms are imposed, then the associated cycle \({\mathcal Z}\) is supported in the special fiber, but need not have dimension \(0!\) In fact, it is sometimes possible to impose 4 independent special endomorphisms and still have a nonempty locus, which may have dimension 0 or 1.
It turns out that the case where \(p\) splits in the real quadratic field differs radically from the case where \(p\) is inert. The first case is similar to the case of modular curves, and in this case a special cycle \({\mathcal Z}\) cut out by three independent special endomorphisms consists of a finite number of points. The more interesting case is when \(p\) is inert. In this case, we determine the precise conditions which ensure that \({\mathcal Z}\) consists of a finite number of points.
For either type of \(p\), when \({\mathcal Z}\) is an Artin scheme, we show that its length coincides with the derivative at \(s=0\), the center of symmetry, of a Fourier coefficient of a Siegel Eisenstein series of genus 3 (Theorems 7.3 and 11.5).

MSC:

11G18 Arithmetic aspects of modular and Shimura varieties
11F30 Fourier coefficients of automorphic forms
11G50 Heights
14G35 Modular and Shimura varieties
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