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

Soumya Das 1  and Jyoti Sengupta 2
  • 1 Department of Mathematics, Indian Institute of Science, Bangalore – 560012, India
  • 2 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai – 400005, India

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 ℚ. More precisely we show that when the L2 norm of an eigenfunction f is one,

fεk12-133+ε
for any ε>0 and for all k sufficiently large.

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