2003/08/29 by Stephen S. Kudla, Kudla, Stephen S.
Mathematics · #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #math.AG #math.NT
paper · pdf · doi:10.48550/arxiv.math/0308296
To appear in the proceedings of the Current Developments in Mathematics seminar held at Harvard University in November of 2002
arxiv created 2003/08/29 · arxiv updated 2009/12/01
This article describes results of joint work with Michael Rapoport and Tonghai Yang. First, we construct an modular form ϕ(τ) 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 t1 and t2 with t1t2 not a square, the (t1,t2)-Fourier coefficient of the height pairing <ϕ(τ1),ϕ(τ2)> coincides with the (t1,t2)-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)≠0, 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)≠0. Detailed proofs will be given elsewhere.