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Derivatives of Eisenstein series and generating functions for arithmetic cycles. (English) Zbl 1066.11026
Séminaire Bourbaki. Volume 1999/2000. Exposés 865–879. Paris: Société Mathématique de France. Astérisque 276, 341-368, Exp. No. 876 (2002).
Summary: In their classic work, F. Hirzebruch and D. B. Zagier [Invent. Math. 36, 57–113 (1976; Zbl 0332.14009)] showed that certain generating functions whose coefficients are the cohomology classes of curves on Hilbert modular surfaces are the \(q\)-expansions of elliptic modular forms of weight 2. In this talk I will describe an analogous family of generating functions whose coefficients arise from arithmetical algebraic geometry, e.g., from \(0\)-cycles on the arithmetic surfaces associated to Shimura curves. The identification of such a function with the derivative of a Siegel-Eisenstein series at its center of symmetry provides a kind of arithmetic analogue of the Siegel-Weil formula.
For the entire collection see [Zbl 0981.00011].

MSC:
11G18 Arithmetic aspects of modular and Shimura varieties
11F30 Fourier coefficients of automorphic forms
11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
14G35 Modular and Shimura varieties
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
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