2014/12/31 by Abhishek Saha
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Character (mathematics) #Conductor #Conjecture #Norm (philosophy) #Upper and lower bounds #math.NT
paper · pdf · doi:10.1093/imrn/rnv259
Final version; to appear in IMRN
arxiv created 2015/09/23 · openalex publication_date 2015/10/01 · arxiv updated 2015/10/16 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We give a lower bound for the sup-norm of an |L2|-normalized newform in an irreducible, unitary, cuspidal representation |π | of |\rm GL2| over a number field. When the central character of |π | is sufficiently ramified, this bound improves upon the trivial bound by a positive power of |N,| where |N| is the norm of the conductor of |π |. This generalizes a result of Templier, who dealt with the special case when the conductor of the central character equals the conductor of the representation. We also make a conjecture about the true size of the sup-norm in the |N|-aspect that takes into account this central character phenomenon. Our results depend upon some explicit formulas and bounds for the Whittaker newvector over a non-archimedean local field, which may be of independent interest.