{
  "id": "1512.05032",
  "version": "v1",
  "published": "2015-12-16T02:41:37.000Z",
  "updated": "2015-12-16T02:41:37.000Z",
  "title": "Generalized Heegner cycles at Eisenstein primes and the Katz $p$-adic $L$-function",
  "authors": [
    "Daniel Kriz"
  ],
  "comment": "53 pages, accepted for publication in Algebra and Number Theory",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper, we consider normalized newforms $f\\in S_k(\\Gamma_0(N),\\varepsilon_f)$ whose non-constant term Fourier coefficients are congruent to those of an Eisenstein series modulo some prime ideal above a rational prime $p$. In this situation, we establish a congruence between the anticyclotomic $p$-adic $L$-function of Bertolini-Darmon-Prasanna and the Katz two-variable $p$-adic $L$-function. From this, we derive congruences between images under the $p$-adic Abel-Jacobi map of certain generalized Heegner cycles attached to $f$ and special values of the Katz $p$-adic $L$-function. In particular, our results apply to newforms associated with elliptic curves $E/\\mathbb{Q}$ whose mod $p$ Galois representations $E[p]$ are reducible at a good prime $p$. As a consequence, we show the following: if $K$ is an imaginary quadratic field satisfying the Heegner hypothesis with respect to $E$ and in which $p$ splits, and if the bad primes of $E$ satisfy certain congruence conditions mod $p$ and $p$ does not divide certain Bernoulli numbers, then the Heegner point $P_{E}(K)$ is non-torsion, in particular implying that $\\text{rank}_{\\mathbb{Z}}E(K) = 1$. From this, we show that when $E$ is semistable with reducible mod $3$ Galois representation, then a positive proportion of real quadratic twists of $E$ have rank 1 and a positive proportion of imaginary quadratic twists of $E$ have rank 0.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-12-16T02:41:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05",
      "11G10",
      "11G40",
      "11G18"
    ],
    "keywords": [
      "generalized heegner cycles",
      "eisenstein primes",
      "galois representation",
      "non-constant term fourier coefficients",
      "real quadratic twists"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 53,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151205032K"
    }
  }
}