{
  "id": "math/0304234",
  "version": "v1",
  "published": "2003-04-16T23:18:15.000Z",
  "updated": "2003-04-16T23:18:15.000Z",
  "title": "Derivatives of Eisenstein series and arithmetic geometry",
  "authors": [
    "Stephen S. Kudla"
  ],
  "journal": "Proceedings of the ICM, Beijing 2002, vol. 2, 173--184",
  "categories": [
    "math.NT"
  ],
  "abstract": "We describe connections between the Fourier coefficients of derivatives of Eisenstein series and invariants from the arithmetic geometry of the Shimura varieties $M$ associated to rational quadratic forms $(V,Q)$ of signature $(n,2)$. In the case $n=1$, we define generating series $\\hat\\phi_1(\\tau)$ for 1-cycles (resp. $\\hat\\phi_2(\\tau)$ for 0-cycles) on the arithmetic surface $\\Cal M$ associated to a Shimura curve over $\\Bbb Q$. These series are related to the second term in the Laurent expansion of an Eisenstein series of weight $\\frac32$ and genus 1 (resp. genus 2) at the Siegel--Weil point, and these relations can be seen as examples of an `arithmetic' Siegel--Weil formula. Some partial results and conjectures for higher dimensional cases are also discussed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2003-04-16T23:18:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14G40",
      "14G35",
      "11F30"
    ],
    "keywords": [
      "eisenstein series",
      "arithmetic geometry",
      "derivatives",
      "higher dimensional cases",
      "rational quadratic forms"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......4234K"
    }
  }
}