{
  "id": "2309.16667",
  "version": "v1",
  "published": "2023-09-28T17:59:22.000Z",
  "updated": "2023-09-28T17:59:22.000Z",
  "title": "Subconvexity for $L$-functions on ${\\rm U}(n) \\times {\\rm U}(n+1)$ in the depth aspect",
  "authors": [
    "Simon Marshall"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $E/F$ be a CM extension of number fields, and let $H < G$ be a unitary Gan--Gross--Prasad pair defined with respect to $E/F$ that is compact at infinity. We consider a family $\\mathcal{F}$ of automorphic representations of $G \\times H$ that is varying at a finite place $w$ that splits in $E/F$. We assume that the representations in $\\mathcal{F}$ satisfy certain conditions, including being tempered and distinguished by the GGP period. For a representation $\\pi \\times \\pi_H \\in \\mathcal{F}$ with base change $\\Pi \\times \\Pi_H$ to ${\\rm GL}_{n+1}(E) \\times {\\rm GL}_n(E)$, we prove a subconvex bound \\[ L(1/2, \\Pi \\times \\Pi_H^\\vee) \\ll C(\\Pi \\times \\Pi_H^\\vee)^{1/4 - \\delta} \\] for any $\\delta < \\tfrac{1}{4n(n+1)(2n^2 + 3n + 3)}$. Our proof uses the unitary Ichino--Ikeda period formula to relate the central $L$-value to an automorphic period, before bounding that period using the amplification method of Iwaniec--Sarnak.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-28T17:59:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F67"
    ],
    "keywords": [
      "depth aspect",
      "unitary ichino-ikeda period formula",
      "subconvexity",
      "unitary gan-gross-prasad pair",
      "automorphic representations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}