JournalsjemsVol. 24, No. 5pp. 1471–1541

Zeros of Rankin–Selberg LL-functions at the edge of the critical strip (with an appendix by Colin J. Bushnell and Guy Henniart)

  • Farrell Brumley

    Université Sorbonne Paris Nord, Villetaneuse, France
  • Jesse Thorner

    University of Illinois, Urbana, USA
  • Asif Zaman

    University of Toronto, Canada
Zeros of Rankin–Selberg $L$-functions at the edge of the critical strip (with an appendix by Colin J. Bushnell and Guy Henniart) cover
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Abstract

Let π\pi (respectively π0\pi_0) be a unitary cuspidal automorphic representation of GLm\mathrm{GL}_m (respectively GLm0\mathrm{GL}_{m_0}) over Q\mathbb{Q}. We prove log-free zero density estimates for Rankin–Selberg LL-functions of the form L(s,π×π0)L(s,\pi\times\pi_0), where π\pi varies in a given family and π0\pi_0 is fixed. These estimates are unconditional in many cases of interest; they hold in full generality assuming an average form of the generalized Ramanujan conjecture. We consider applications of these estimates related to mass equidistribution for Hecke–Maaß forms, the rarity of Landau–Siegel zeros of Rankin–Selberg LL-functions, the Chebotarev density theorem, and \ell-torsion in class groups of number fields.

Cite this article

Farrell Brumley, Jesse Thorner, Asif Zaman, Zeros of Rankin–Selberg LL-functions at the edge of the critical strip (with an appendix by Colin J. Bushnell and Guy Henniart). J. Eur. Math. Soc. 24 (2022), no. 5, pp. 1471–1541

DOI 10.4171/JEMS/1134