Current Developments in Mathematics

Modular forms and arithmetic geometry

Stephen S. Kudla

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The aim of these notes is to describe some examples of modular forms whose Fourier coefficients involve quantities from arithmeticla algebraic geometry. Ath the moment, no general theory of such forms exists, but the examples suggest that they should be viewed as a kind of arithmetic analogue of theta series and that there should be an arithmetic Siegel-Weil formula relating suitable averages of them to special values of derivatives of Eisenstein series. We will concentrate on the case for which the most complete picture is available, the case of generating series for cycles on the arithmetic surfaces associated to Shimura curves over ?, expanding on the treatment in [40]. A more speculative overview can be found in [41].

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D. Jerison, B. Mazur, T. Mrowka, W. Schmid, R. Stanley, S-T. Yau, eds. Current Developments in Mathematics, 2002 (Boston: International Press, 2003), 135-179

First available in Project Euclid: 29 June 2004

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Kudla, Stephen S. Modular forms and arithmetic geometry. Current Developments in Mathematics, 2002, 135--179, International Press of Boston, Boston, MA, 2003.

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