Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Modular forms and arithmetic geometry   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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The categories of this paper are: physics
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The categories of this paper are: acc-phys
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The categories of this paper are: plasm-ph
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The categories of this paper are: q-bio
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The categories of this paper are: ao-sci
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The categories of this paper are: q-fin
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The categories of this paper are: hep-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-th
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The categories of this paper are: eess
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The categories of this paper are: chem-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: quant-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: astro-ph