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

Abstract

Abstract We prove a sub-convex estimate for the sup-norm of L 2-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 L 2 norm of an eigenfunction f is one, ∥ f ∥ ∞ ≪ ε k 1 2 - 1 33 + ε $ \Vert f \Vert _\infty \ll _\varepsilon k^{\frac{1}{2} - \frac{1}{33} +\varepsilon } $ for any ε > 0 ${\varepsilon >0}$ and for all k sufficiently large.