Derivatives of Eisenstein series and arithmetic geometry
Abstract
Description
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 $\hatϕ_1(τ)$ for 1-cycles (resp. $\hatϕ_2(τ)$ 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.