L^∞ norms of holomorphic modular forms in the case of compact quotient

2013/01/16 by Das, Soumya, Sengupta, Jyoti
#11F11 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1301.3677

Abstract

We prove a sub-convex estimate for the sup-norm of L²-normalized holomorphic modular forms of weight k on the upper half plane, with respect to the unit group of a quaternion division algebra over \mf Q. More precisely we show that when the L² norm of an eigenfunction f is one, | f |_∞ ≪ k^{1/2 - 12/131 + ε}, for any ε>0 and for all k sufficiently large.

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