{
  "id": "2007.09803",
  "version": "v1",
  "published": "2020-07-19T22:31:25.000Z",
  "updated": "2020-07-19T22:31:25.000Z",
  "title": "On $L$-Functions of Modular Elliptic Curves and Certain $K3$ Surfaces",
  "authors": [
    "Malik Amir",
    "Letong",
    "Hong"
  ],
  "comment": "18 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Inspired by Lehmer's conjecture on the nonvanishing of the Ramanujan $\\tau$-function, one may ask whether an odd integer $\\alpha$ can be equal to $\\tau(n)$ or any coefficient of a newform $f(z)$. Balakrishnan, Craig, Ono, and Tsai used the theory of Lucas sequences and Diophantine analysis to characterize non-admissible values of newforms of even weight $k\\geq 4$. We use these methods for weight $2$ and $3$ newforms and apply our results to $L$-functions of modular elliptic curves and certain $K3$ surfaces with Picard number $\\ge 19$. In particular, for the complete list of weight $3$ newforms $f_\\lambda(z)=\\sum a_\\lambda(n)q^n$ that are $\\eta$-products, and for $N_\\lambda$ the conductor of some elliptic curve $E_\\lambda$, we show that if $|a_\\lambda(n)|<100$ is odd with $n>1$ and $(n,2N_\\lambda)=1$, then \\begin{align*} a_\\lambda(n) \\in \\,& \\{-5,9,\\pm 11,25, \\pm41, \\pm 43, -45,\\pm47,49, \\pm53,55, \\pm59, \\pm61, \\pm 67\\}\\\\ & \\,\\,\\, \\cup \\, \\{-69,\\pm 71, \\pm 73,75, \\pm79,\\pm81, \\pm 83, \\pm89,\\pm 93 \\pm 97, 99\\}. \\end{align*} Assuming the Generalized Riemann Hypothesis, we can rule out a few more possibilities leaving \\begin{align*} a_\\lambda(n) \\in \\{-5,9,\\pm 11,25,-45,49,55,-69,75,\\pm 81,\\pm 93, 99\\}. \\end{align*}",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-19T22:31:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "modular elliptic curves",
      "diophantine analysis",
      "odd integer",
      "picard number",
      "complete list"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}