{
  "id": "2001.04310",
  "version": "v1",
  "published": "2020-01-13T14:46:29.000Z",
  "updated": "2020-01-13T14:46:29.000Z",
  "title": "On Shafarevich-Tate groups and analytic ranks in Hida families of modular forms",
  "authors": [
    "Stefano Vigni"
  ],
  "comment": "34 pages",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Let $f$ be a newform of weight $2$, square-free level and trivial character, let $A_f$ be the abelian variety attached to $f$ and for every good ordinary prime $p$ for $f$ let $\\boldsymbol f^{(p)}$ be the $p$-adic Hida family through $f$. We prove that, for all but finitely many primes $p$ as above, if $A_f$ is an elliptic curve such that $A_f(\\mathbb Q)$ has rank $1$ and the $p$-primary part of the Shafarevich-Tate group of $A_f$ over $\\mathbb Q$ is finite then all 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 $1$-dimensional image of the relevant $p$-adic \\'etale Abel-Jacobi map. Analogous results are obtained also in the rank $0$ case. As a second contribution, with no restriction on the dimension of $A_f$ but assuming the non-degeneracy of certain height pairings \\`a la Gillet-Soul\\'e between Heegner cycles, we show that if $f$ has analytic rank $1$ then, for all but finitely many $p$, all specializations of $\\boldsymbol f^{(p)}$ of weight congruent to $2$ modulo $2(p-1)$ and trivial character have analytic rank $1$. This result provides some evidence in rank $1$ and weight larger than $2$ for a conjecture of Greenberg predicting that the analytic ranks of even weight modular forms in a Hida family should be as small as allowed by the functional equation, with at most finitely many exceptions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-13T14:46:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F11",
      "14C25"
    ],
    "keywords": [
      "analytic rank",
      "shafarevich-tate group",
      "hida family",
      "trivial character",
      "weight congruent"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}