{
  "id": "1805.10361",
  "version": "v1",
  "published": "2018-05-25T20:49:15.000Z",
  "updated": "2018-05-25T20:49:15.000Z",
  "title": "On the number of Galois orbits of newforms",
  "authors": [
    "Luis Dieulefait",
    "Ariel Pacetti",
    "Panagiotis Tsaknias"
  ],
  "comment": "26 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Counting the number of Galois orbits of newforms in $S_k(\\Gamma_0(N))$ and giving some arithmetic sense to this number is an interesting open problem. The case $N=1$ corresponds to Maeda's conjecture (still an open problem) and the expected number of orbits in this case is 1, for any $k \\ge 16$. In this article we give local invariants of Galois orbits of newforms for general $N$ and count their number. Using an existence result of newforms with prescribed local invariants we prove a lower bound for the number of non-CM Galois orbits of newforms for $\\Gamma_0(N)$ for large enough weight $k$ (under some technical assumptions on $N$). Numerical evidence suggests that in most cases this lower bound is indeed an equality, thus we leave as a Question the possibility that a generalization of Maeda's conjecture could follow from our work. We finish the paper with some natural generalizations of the problem and show some of the implications that a generalization of Maeda's conjecture has.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-25T20:49:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F03",
      "11F11"
    ],
    "keywords": [
      "maedas conjecture",
      "lower bound",
      "local invariants",
      "non-cm galois orbits",
      "interesting open problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}