{
  "id": "1506.08723",
  "version": "v1",
  "published": "2015-06-29T16:38:18.000Z",
  "updated": "2015-06-29T16:38:18.000Z",
  "title": "Nonvanishing of central values of $L$-functions of newforms in $S_2 (Γ_0 (dp^2))$ twisted by quadratic characters",
  "authors": [
    "Samuel Le Fourn"
  ],
  "comment": "14 pages, comments very welcome",
  "categories": [
    "math.NT"
  ],
  "abstract": "We prove that for $d \\in \\{ 2,3,5,7,13 \\}$ and a quadratic real field $K$ (or $\\Q$) of discriminant $D$ and Dirichlet character $\\chi$, if a prime $p$ is large enough compared to $D$, there is an eigenform $f \\in S_2(\\Gamma_0(dp^2))$ with sign $(+1)$ with respect to the Atkin-Lehner involution $w_{p^2}$ such that $L(f \\otimes \\chi,1) \\neq 0$. This result is obtained through an estimate of a weighted sum of twists of $L$-functions which generalises a result of Ellenberg et al., and relies on classical techniques and a Petersson trace formula restricted to Atkin-Lehner eigenspaces. This has applications in the study of rational points on some families of modular curves.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-06-29T16:38:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quadratic characters",
      "central values",
      "quadratic real field",
      "petersson trace formula",
      "dirichlet character"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150608723L"
    }
  }
}