{
  "id": "1012.3513",
  "version": "v1",
  "published": "2010-12-16T04:50:12.000Z",
  "updated": "2010-12-16T04:50:12.000Z",
  "title": "Graphs of Hecke operators",
  "authors": [
    "Oliver Lorscheid"
  ],
  "comment": "36 pages",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Let $X$ be a curve over $\\F_q$ with function field $F$. In this paper, we define a graph for each Hecke operator with fixed ramification. A priori, these graphs can be seen as a convenient language to organize formulas for the action of Hecke operators on automorphic forms. However, they will prove to be a powerful tool for explicit calculations and proofs of finite dimensionality results. We develop a structure theory for certain graphs $G_x$ of unramified Hecke operators, which is of a similar vein to Serre's theory of quotients of Bruhat Tits trees. To be precise, $G_x$ is locally a quotient of a Bruhat Tits tree and has finitely many components. An interpretation of $G_x$ in terms of rank 2 bundles on $X$ and methods from reduction theory show that $G_x$ is the union of finitely many cusps, which are infinite subgraphs of a simple nature, and a nucleus, which is a finite subgraph that depends heavily on the arithmetics of $F$. We describe how one recovers unramified automorphic forms as functions on the graphs $G_x$. In the exemplary cases of the cuspidal and the toroidal condition, we show how a linear condition on functions on $G_x$ leads to a finite dimensionality result. In particular, we re-obtain the finite-dimensionality of the space of unramified cusp forms and the space of unramified toroidal automorphic forms. In an Appendix, we calculate a variety of examples of graphs over rational function fields.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-12-16T04:50:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C75",
      "11F41",
      "11G20",
      "11R58",
      "14D24",
      "14H05",
      "14H60",
      "20C08"
    ],
    "keywords": [
      "hecke operator",
      "bruhat tits tree",
      "finite dimensionality result",
      "unramified toroidal automorphic forms",
      "rational function fields"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1012.3513L"
    }
  }
}