On the $L^6$-norm of holomorphic Hecke eigenforms

Abstract

Let $H_k$ be an $L^2$-normalized Hecke basis for the space of all holomorphic cusp forms of weight $k$. We show that ${\max }_{f\in H_k}\Vert F{\Vert }_6\gg (\log \log k{)}^{\frac{1}{2}}$ where $F(z)=(\Im z{)}^{\frac{k}{2}}f(z).$ This confirms that the $L^6$-norm of Hecke eigenforms does not converge uniformly as the weight goes to infinity. We also give some results on the joint mass of degree $6$.

7 pages. Comments welcome!