2018/12/04 by Abhishek Saha, Saha, Abhishek · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Analytic Number Theory Research
paper · pdf · doi:10.48550/arxiv.1812.01572
Let D be an indefinite quaternion division algebra over \ℚ. We\napproach the problem of bounding the sup-norms of automorphic forms \φ on\nD^\×( mathbbA) that belong to irreducible automorphic representations\nand transform via characters of unit groups of orders of D. We obtain a\nnon-trivial upper bound for \‖\φ\‖_\∞ in the level aspect that is valid\nfor arbitrary orders. This generalizes and strengthens previously known upper\nbounds for \‖\φ\‖_\∞ in the setting of newforms for Eichler orders. In\nthe special case when the index of the order in a maximal order is a squarefull\ninteger N, our result specializes to \‖\φ\‖_\∞ \≪\π_\∞,\n\ε N1/3 + \ε \‖\φ\‖2.\n A key application of our result is to automorphic forms \φ which\ncorrespond at the ramified primes to either minimal vectors (in the sense of\nHu-Nelson-Saha), or p-adic microlocal lifts (in the sense of Nelson). For\nsuch forms, our bound specializes to \‖ \φ\‖_\∞ \≪\ε\nC frac16 + \ε\‖\φ\‖2 where C is the conductor of the\nrepresentation \π generated by \φ. This improves upon the previously\nknown local bound \‖\φ\‖_\∞ \≪\λ, \ε C frac14 +\n\ε\‖\φ\‖2 in these cases.\n