Derivatives of Eisenstein series and arithmetic geometry

Details Derivatives of Eisenstein series and arithmetic geometry Derivatives of Eisenstein series and arithmetic geometry

2003-04-16

arxiv.org/abs/math/0304234v1

Proceedings of the ICM, Beijing 2002, vol. 2, 173--184

Mathematics

Number Theory

Scientific paper

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.

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