{
  "id": "1701.03189",
  "version": "v1",
  "published": "2017-01-11T23:49:12.000Z",
  "updated": "2017-01-11T23:49:12.000Z",
  "title": "Identities between Hecke Eigenforms",
  "authors": [
    "Dianbin Bao"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper, we study solutions to $h=af^2+bfg+g^2$, where $f,g,h$ are Hecke newforms with respect to $\\Gamma_1(N)$ of weight $k>2$ and $a,b\\neq 0$. We show that the number of solutions is finite for all $N$. Assuming Maeda's conjecture, we prove that the Petersson inner product $\\langle f^2,g\\rangle$ is nonzero, where $f$ and $g$ are any nonzero cusp eigenforms for $SL_2(\\mathbb{Z})$ of weight $k$ and $2k$, respectively. As a corollary, we obtain that, assuming Maeda's conjecture, identities between cusp eigenforms for $SL_2(\\mathbb{Z})$ of the form $X^2+\\sum_{i=1}^n \\alpha_iY_i=0$ all are forced by dimension considerations. We also give a proof using polynomial identities between eigenforms that the $j$-function is algebraic on zeros of Eisenstein series of weight $12k$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-11T23:49:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hecke eigenforms",
      "assuming maedas conjecture",
      "petersson inner product",
      "nonzero cusp eigenforms",
      "eisenstein series"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}