{
  "id": "2112.11847",
  "version": "v2",
  "published": "2021-12-22T12:49:16.000Z",
  "updated": "2023-05-09T09:18:53.000Z",
  "title": "On Shafarevich-Tate groups and analytic ranks in families of modular forms, II. Coleman families",
  "authors": [
    "Maria Rosaria Pati",
    "Gautier Ponsinet",
    "Stefano Vigni"
  ],
  "comment": "Slight revision following the referee's report; 32 pages. Final version, to appear in Mathematical Research Letters",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "This is the second article in a two-part project whose aim is to study algebraic and analytic ranks in $p$-adic families of modular forms. Let $f$ be a newform of weight $2$, square-free level $N$ and trivial character, let $A_f$ be the abelian variety attached to $f$, whose dimension will be denoted by $d_f$, and for every prime number $p\\nmid N$ let $\\boldsymbol f^{(p)}$ be a $p$-adic Coleman family through $f$ over a suitable open disc in the $p$-adic weight space. We prove that, for all but finitely many primes $p$ as above, if $A_f(\\mathbb Q)$ has rank $r\\in\\{0,d_f\\}$ and the $p$-primary part of the Shafarevich-Tate group of $A_f$ over $\\mathbb Q$ is finite, then all classical specializations of $\\boldsymbol f^{(p)}$ of weight congruent to $2$ modulo $2(p-1)$ and trivial character have finite $p$-primary Shafarevich-Tate group and $r/d_f$-dimensional image of the relevant $p$-adic \\'etale Abel-Jacobi map. As a second contribution, assuming the non-degeneracy of certain height pairings \\`a la Gillet-Soul\\'e between Heegner cycles, we show that, for all but finitely many $p$, if $f$ has analytic rank $r\\in\\{0,1\\}$, then all classical specializations of $\\boldsymbol f^{(p)}$ of weight congruent to $2$ modulo $2(p-1)$ and trivial character have analytic rank $r$. This result provides some evidence for a conjecture of Greenberg on analytic ranks in families of modular forms.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2023-05-09T09:18:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F11",
      "14C25"
    ],
    "keywords": [
      "analytic rank",
      "modular forms",
      "coleman family",
      "trivial character",
      "weight congruent"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}