Derivatives of Eisenstein series and arithmetic geometry

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 ${\widehat{\varphi }}_1(\tau )$ for 1-cycles (resp. ${\widehat{\varphi }}_2(\tau )$ for 0-cycles) on the arithmetic surface $\text{Cal}M$ associated to a Shimura curve over $\mathbb{Q}$. These series are related to the second term in the Laurent expansion of an Eisenstein series of weight $\frac{3}{2}$ 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.