{
  "id": "1802.04419",
  "version": "v1",
  "published": "2018-02-13T01:19:53.000Z",
  "updated": "2018-02-13T01:19:53.000Z",
  "title": "Iwasawa theory for Rankin--Selberg products of $p$-non-ordinary eigenforms",
  "authors": [
    "Kazim Büyükboduk",
    "Antonio Lei",
    "David Loeffler",
    "Guhan Venkat"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $f$ and $g$ be two modular forms which are non-ordinary at $p$. The theory of Beilinson-Flach elements gives rise to four rank-one non-integral Euler systems for the Rankin-Selberg convolution $f \\otimes g$, one for each choice of $p$-stabilisations of $f$ and $g$. We prove (modulo a hypothesis on non-vanishing of $p$-adic $L$-fuctions) that the $p$-parts of these four objects arise as the images under appropriate projection maps of a single class in the wedge square of Iwasawa cohomology, confirming a conjecture of Lei-Loeffler-Zerbes. Furthermore, we define an explicit logarithmic matrix using the theory of Wach modules, and show that this describes the growth of the Euler systems and $p$-adic $L$-functions associated to $f \\otimes g$ in the cyclotomic tower. This allows us to formulate \"signed\" Iwasawa main conjectures for $f\\otimes g$ in the spirit of Kobayashi's $\\pm$-Iwasawa theory for supersingular elliptic curves; and we prove one inclusion in these conjectures under our running hypotheses.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-02-13T01:19:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R23",
      "11F11",
      "11R20"
    ],
    "keywords": [
      "iwasawa theory",
      "rankin-selberg products",
      "non-ordinary eigenforms",
      "rank-one non-integral euler systems",
      "appropriate projection maps"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}