7 problems

Let $S_{\kappa}$ be the space of cusp forms of weight $\kappa$, and let $[E_k,E_\ell]_n$ denote the $n$-th Rankin–Cohen bracket of Eisenstein series of weights $k$ and $\ell$, wher…

Monotonicity conjecture. For all $\theta\in\left[\frac{\pi}{2},1.9\right]$, the first function is decreasing on $[0,0.5]$, and for all $\theta\in\left[1.9,\frac{2\pi}{3}\right]$, t…

Let $S_{j,2}(Gamma_2,epsilon)$ denote the space of Siegel cusp forms of degree $2$, weight $(j,2)$, and character $epsilon$. The character-valued vanishing conjecture states that ……

Square-root cancellation conjecture. As $N\to\infty$,

Let … The form $F^{(n)}$ is a Hermitian modular form of weight $12$; in particular, the theorem establishes that $F^{(4)}\not\equiv 0$ and that either $F^{(3)}\equiv 0$ or…

Let $f$ be a cusp form with archimedean parameter $t_f$, and let $\\|f\\|_2$ and $\\|f\\|_{\infty}$ denote its $L^2$- and sup-norms, respectively. Sup-norm conjecture. One might co…

Let $GL(3,\mathbb{Z})\backslash GL(3,\mathbb{R})$ be the full-level quotient, and consider cohomological cusp forms on it, apart from the known symmetric square lifts from $GL(2)$.…