Abstract: A series of conjectures of Steve Kudla, following classical work of Hirzerbuch-Zagier, Gross-Zagier and Gross-Kohnen-Zagier, predicts that (arithmetic) special cycle classes on certain Shimura varieties can be viewed as Fourier coefficients of automorphic forms in a suitable sense. There is a difficulty even in the formulation of the problem, because the 'natural' definition of cycles associated with degenerate coefficients yields cycles in the wrong degree, an issue that gets rather more severe when one tries to work over the integral model. I'll explain how the theory of virtual fundamental classes for certain derived complete intersection maps gives a systematic fix for this issue, and how recent developments in integral p-adic Hodge theory (joint w. Mathew and Gardner) help to construct such maps.