The principal investigator is studying generalizations of the Gross-Zagier theorem, relating intersection multiplicities of special cycles on Shimura varieties to central values and central derivatives ofautomorphic $L$-functions. The original theorem of Gross and Zagier expresses the arithmetic intersections of complex multiplication points on modular curves in terms of Fourier coefficients of a certain explicit modular form. Gross and Zagier use this to deduce a relation between the Neron-Tate heights of Heegner points on modular Jacobians and derivatives of L-functions. Results and conjectures of Borcherds, Gross-Kudla, Hirzebruch-Zagier, Kudla-Rappoport-Yang, and Zhang suggest that the Gross-Zagier theorem is merely the simplest cases of a much broader theory relating arithmetic intersections of special cycles on Shimura varieties to Fourier coefficients of modular forms. Such a theory would yield results toward generalized forms of the Birch and Swinnerton-Dyer conjecture, e.g. the Bloch-Kato conjectures. Toward this end, the principal investigator is studying two generalized forms of the Gross-Zagier theorem. The first project is to extend the original Gross-Zagier theorem to include intersections of special points on modular curves with additional level structure. Such a result would yield new cases of the Birch and Swinnerton-Dyer conjecture for abelian varieties attached to modular forms with nontrivial nebentype. The second project is a part of a vast series of conjectures of Kudla concerning the arithmetic intersections of special cycles on Shimura varieties of orthogonal type. The case of interest to the principal investigator involves the computation of intersection multiplicities on a class of Shimura surfaces which includes the classical Hilbert modular surfaces, and the comparison of these intersection multiplicities with Fourier coefficients of automorphic forms.
In the field of arithmetic geometry certain there are certain curves, surfaces, and higher dimension analogs which play a central role. These objects are called Shimura varieties, and are interesting at least in part because they encode arithmetic information (i.e. properties of the integers and rational numbers) in a geometric form. These Shimura varieties contain inside them many interesting objects of lower dimensions. For example the one-dimensional Shimura varieties come equipped with a family of special points, the two-dimensional Shimura varieties come equipped with both special points and special curves on the surface, three-dimensional Shimura varieties have special points, curves, and surfaces inside them, and so on. One way in which the geometry of these objects encodes arithmetic information is through ntersection theory. If, for example, one takes a Shimura surface and two special curves lying on the surface, then one may simply count the number of times that the two curves intersect one another. Work of Hirzebruch and Zagier, dating back to the 1970's, shows that these geometrically defined intersection numbers agree with sequences of numbers arising in arithmetic. This connection between geometry and arithmetic was later exploited by Gross and Zagier to prove fundamental results about elliptic curves, objects of great importance both in pure math (e.g. to the proof of Fermat's last theorem) and in cryptography. The principal investigator is working to extend some of the theory to higher dimensions by computing the intersection numbers of a surface with a family of curves, all inside of a three-dimensional Shimura variety, and comparing these with numbers arising from arithmetic. The principal investigator expects that this will lead to proofs of special cases of some long-standing and important conjectures in number theory.
