The Jacobi newform sup-norm conjecture

Let $J^{cusp}_{k,1}(N)$ denote the space of Jacobi cusp forms of index $1$ and level $N$, and let $\phi\in J^{cusp}_{k,1}(N)$ be an $L^2$-normalized newform. Write $\lVert\phi\rVert_\infty$ for its supremum norm. Jacobi newform sup-norm conjecture. Let $k \ge 4$ be even and $\phi\in J^{cusp}_{k,1}(N)$ be an $L^2$-normalized newform. Then for large $N$ and any $\epsilon>0$, one has

$$\lVert\phi\rVert_\infty \ll_\epsilon N^{-1/2+\epsilon}.$$

The paper proves a nontrivial individual bound, but presents this stronger level-aspect estimate as a conjecture; it is intended as the expected trivial-scale bound for a Jacobi newform.