{
  "id": "2205.09527",
  "version": "v1",
  "published": "2022-05-19T12:50:00.000Z",
  "updated": "2022-05-19T12:50:00.000Z",
  "title": "Average analytic ranks of elliptic curves over number fields",
  "authors": [
    "Tristan Phillips"
  ],
  "comment": "25 pages. Split from arXiv:2201.10624. Comments welcome!",
  "categories": [
    "math.NT"
  ],
  "abstract": "We give a conditional bound for the average analytic rank of elliptic curves over an arbitrary number field. In particular, under the assumptions that all elliptic curves over a number field $K$ are modular and have $L$-functions which satisfy the Generalized Riemann Hypothesis, we show that the average analytic rank of isomorphism classes of elliptic curves over $K$ is bounded above by $3\\text{deg}(K)+1/2$. A key ingredient in the proof of this result is to give asymptotics for the number of elliptic curves over an arbitrary number field with a prescribed local condition; these results will follow from general results for counting points of bounded height on weighted projective spaces with a prescribed local condition.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-19T12:50:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05",
      "11D45",
      "11G07",
      "11G35",
      "11G40",
      "11G50",
      "14G05"
    ],
    "keywords": [
      "average analytic rank",
      "elliptic curves",
      "arbitrary number field",
      "prescribed local condition",
      "isomorphism classes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}