{
  "id": "math/0308296",
  "version": "v1",
  "published": "2003-08-29T17:55:01.000Z",
  "updated": "2003-08-29T17:55:01.000Z",
  "title": "Modular forms and arithmetic geometry",
  "authors": [
    "Stephen S. Kudla"
  ],
  "comment": "To appear in the proceedings of the Current Developments in Mathematics seminar held at Harvard University in November of 2002",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2003-08-29T17:55:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "modular form",
      "arithmetic geometry",
      "arithmetic chow group",
      "arithmetic theta lift",
      "local obstruction"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......8296K"
    }
  }
}