Modular forms and arithmetic geometry

Details Modular forms and arithmetic geometry Modular forms and arithmetic geometry

2003-08-29

arxiv.org/abs/math/0308296v1

Mathematics

Number Theory

To appear in the proceedings of the Current Developments in Mathematics seminar held at Harvard University in November of 2002

Scientific paper

This article describes results of joint work with Michael Rapoport and Tonghai Yang. First, we construct an modular form \phi(\tau) of weight 3/2 valued in the arithmetic Chow group of the arithmetic surface M attached toa Shimura curve over Q. The q-expansion of this function is an analogue of the Hirzebruch-Zagier generating function for the cohomology classes of curves on a Hilbert modular surface. This`arithmetic theta function' is used to define an `arithmetic theta lift' from modular forms of weight 3/2 to the arithmetic Chow group of M. For integers t_1 and t_2 with t_1t_2 not a square, the (t_1,t_2)-Fourier coefficient of the height pairing <\phi(\tau_1),\phi(\tau_2)> coincides with the (t_1,t_2)-Fourier coefficient of the restriction to the diagonal of the central derivative of a certain Eisenstein series of weight 3/2 and genus 2. Using this fact and results about the doubling integral for forms of weight 3/2, we prove that the arithmetic theta lift of a Hecke eigenform f is nonzero if and only if there is no local obstruction (theta dichotomy) and the standard Hecke L-function L(s,F) of the corresponding newform F of weight 2 has nonvanishing derivative, L'(1,F)\ne0, at the center of symmetry. This is an analogue of a result of Waldspurger according to which the classical Shimura lift of such a form is nonzero if and only if there is no local obstruction and L(1,F)\ne0. Detailed proofs will be given elsewhere.

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