  We describe connections between the Fourier coefficients of derivatives of
Eisenstein series and invariants from the arithmetic geometry of the Shimura
varieties $M$ associated to rational quadratic forms $(V,Q)$ of signature
$(n,2)$. In the case $n=1$, we define generating series $\hat\phi_1(\tau)$ for
1-cycles (resp. $\hat\phi_2(\tau)$ for 0-cycles) on the arithmetic surface
$\Cal M$ associated to a Shimura curve over $\Bbb Q$. These series are related
to the second term in the Laurent expansion of an Eisenstein series of weight
$\frac32$ and genus 1 (resp. genus 2) at the Siegel--Weil point, and these
relations can be seen as examples of an `arithmetic' Siegel--Weil formula. Some
partial results and conjectures for higher dimensional cases are also
discussed.
