The Arithmetic and Geometry of Integral Models of Orthogonal Shimura Varieties

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The main purpose of this project is to tease out certain hidden identities between mathematical objects of very disparate origins: One defined using the geometry of certain spaces defined by polynomial equations, and one defined using infinite sums of functions with complex values. Such identities have proven to have fruitful applications to elliptic curve cryptography. The first instance of such a formula was discovered by the mathematicians Gross and Zagier in the 1980s, who worked with one dimensional spaces. The PI will extend their work to higher dimensions.Shimura varieties have proven to be fundamental objects in modern day arithmetic geometry and number theory, with applications ranging from Langlands reciprocity to the understanding of abelian varieties and K3 surfaces to instances of the Birch-Swinnerton-Dyer conjecture (through the Gross-Zagier formula). The main purpose of this project is to study a special class of such Shimura varieties, attached to orthogonal groups, which are simple enough to be accessible, yet carry rich arithmetic information. Steve Kudla, along with his collaborators, has formulated a series of wide ranging conjectures on the intersection theory of these varieties, both classical and arithmetic (in the Arakelov sense), relating them to special values and Fourier coefficients of certain L-functions and Eisenstein series (and their derivatives). The key part of this project is concerned with proving some new cases of these conjectures, through the study of integral models of orthogonal Shimura varieties and their properties. Among other things, the eventual results of the work of the PI and his collaborators should yield a proof of a conjectural formula of P. Colmez on the heights of CM abelian varieties.