{
  "id": "2411.03278",
  "version": "v1",
  "published": "2024-11-05T17:20:46.000Z",
  "updated": "2024-11-05T17:20:46.000Z",
  "title": "Distribution of slopes for $\\mathscr{L}$-invariants",
  "authors": [
    "Jiawei An"
  ],
  "comment": "39 pages; comments are welcome",
  "categories": [
    "math.NT"
  ],
  "abstract": "Fix a prime $p\\geq5$, an integer $N\\geq1$ relatively prime to $p$, and an irreducible residual global Galois representation $\\bar{r}: Gal_{\\mathbb{Q}}\\rightarrow GL_2(\\mathbb{F}_p)$. In this paper, we utilize ghost series to study $p$-adic slopes of $\\mathscr{L}$-invariants for $\\bar{r}$-newforms. More precisely, under a locally reducible and strongly generic condition for $\\bar{r}$: (1) we determine the slopes of $\\mathscr{L}$-invariants associated to $\\bar{r}$-newforms of weight $k$ and level $\\Gamma_0(Np)$, with at most $O(log_pk)$ exceptions; (2) we establish the integrality of these slopes; (3) we prove an equidistribution property for these slopes as the weight $k$ tends to infinity, which confirms the equidistribution conjecture for $\\mathscr{L}$-invariants proposed by Bergdall--Pollack recently.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-05T17:20:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F33",
      "11F85"
    ],
    "keywords": [
      "invariants",
      "irreducible residual global galois representation",
      "equidistribution property",
      "strongly generic condition",
      "utilize ghost series"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}