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,
for any and for all k sufficiently large.
Funding source: DST-INSPIRE
Award Identifier / Grant number: IFA 12-MA-13
It is a great pleasure for the authors to thank Professor Peter Sarnak for his thoughtful comments on the paper. We also thank the School of Mathematics TIFR, Mumbai and the Department of Mathematics, IISc., Bangalore, where parts of this work was carried out, for providing excellent working atmosphere. The authors are grateful to the anonymous referee for a careful reading of the paper and for many helpful suggestions that improved the presentation.
© 2015 by De Gruyter